```python
import os
from galgebra_ipython_helpers import check
os.chdir('../LaTeX')
```

```python
check('4Derr')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} g_{ii} = \left[\begin{array}{cccc}0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\end{array}\right] \end{equation*}
    \begin{equation*} x_1\rfloor x_1== 1 \end{equation*}
    \begin{equation*} x_1\rfloor x_2==  0 \end{equation*}
    \begin{equation*} x_2\rfloor x_1==  0 \end{equation*}
    \begin{equation*} x_2\rfloor x_2== 1 \end{equation*}
    $-\infty < v < \infty$
    \begin{equation*} (-v (x_1\W x_2)/2).exp()== \cos{\left (\frac{\left|{v}\right|}{2} \right )}  - \frac{v \sin{\left (\frac{\left|{v}\right|}{2} \right )}}{2 \left|{v}\right|} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} - \frac{v \sin{\left (\frac{\left|{v}\right|}{2} \right )}}{2 \left|{v}\right|} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{4} + \frac{v \sin{\left (\frac{\left|{v}\right|}{2} \right )}}{2 \left|{v}\right|} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} - \frac{v \sin{\left (\frac{\left|{v}\right|}{2} \right )}}{2 \left|{v}\right|} \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{4} \end{equation*}
    $0\le v < \infty$
    \begin{equation*} (-v (x_1\W x_2)/2).exp()== \cos{\left (\frac{v}{2} \right )}  - \frac{\sin{\left (\frac{v}{2} \right )}}{2} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} - \frac{\sin{\left (\frac{v}{2} \right )}}{2} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{4} + \frac{\sin{\left (\frac{v}{2} \right )}}{2} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} - \frac{\sin{\left (\frac{v}{2} \right )}}{2} \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{4} \end{equation*}
    \end{document}

```python
check('curvi_linear_latex')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    Derivatives in Spherical Coordinates
    \begin{equation*} f = f \end{equation*}
    \begin{equation*} A = A^{r}  \boldsymbol{e}_{r} + A^{\theta }  \boldsymbol{e}_{\theta } + A^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} B = B^{r\theta }  \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\theta } + B^{r\phi }  \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\phi } + B^{\theta \phi }  \boldsymbol{e}_{\theta }\wedge \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \partial_{r} f  \boldsymbol{e}_{r} + \frac{\partial_{\theta } f }{r^{2}} \boldsymbol{e}_{\theta } + \frac{\partial_{\phi } f }{r^{2} {\sin{\left (\theta  \right )}}^{2}} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot A = \frac{A^{\theta } }{\tan{\left (\theta  \right )}} + \partial_{\phi } A^{\phi }  + \partial_{r} A^{r}  + \partial_{\theta } A^{\theta }  + \frac{2 A^{r} }{r} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \times A = -I (\boldsymbol{\nabla} \W A) = \left(\frac{2 A^{\phi } }{\tan{\left (\theta  \right )}} + \partial_{\theta } A^{\phi }  - \frac{\partial_{\phi } A^{\theta } }{{\sin{\left (\theta  \right )}}^{2}}\right) \left|{\sin{\left (\theta  \right )}}\right| \boldsymbol{e}_{r} + \frac{- r^{2} {\sin{\left (\theta  \right )}}^{2} \partial_{r} A^{\phi }  - 2 r A^{\phi }  {\sin{\left (\theta  \right )}}^{2} + \partial_{\phi } A^{r} }{r^{2} \left|{\sin{\left (\theta  \right )}}\right|} \boldsymbol{e}_{\theta } + \frac{r^{2} \partial_{r} A^{\theta }  + 2 r A^{\theta }  - \partial_{\theta } A^{r} }{r^{2} \left|{\sin{\left (\theta  \right )}}\right|} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \nabla^{2}f = \frac{r^{2} \partial^{2}_{r} f  + 2 r \partial_{r} f  + \partial^{2}_{\theta } f  + \frac{\partial_{\theta } f }{\tan{\left (\theta  \right )}} + \frac{\partial^{2}_{\phi } f }{{\sin{\left (\theta  \right )}}^{2}}}{r^{2}} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \W B = \frac{r^{2} \partial_{r} B^{\theta \phi }  + 4 r B^{\theta \phi }  - \frac{2 B^{r\phi } }{\tan{\left (\theta  \right )}} - \partial_{\theta } B^{r\phi }  + \frac{\partial_{\phi } B^{r\theta } }{{\sin{\left (\theta  \right )}}^{2}}}{r^{2}} \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\theta }\wedge \boldsymbol{e}_{\phi } \end{equation*}
    Derivatives in Paraboloidal Coordinates
    \begin{equation*} f = f \end{equation*}
    \begin{equation*} A = A^{u}  \boldsymbol{e}_{u} + A^{v}  \boldsymbol{e}_{v} + A^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} B = B^{uv}  \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{v} + B^{u\phi }  \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{\phi } + B^{v\phi }  \boldsymbol{e}_{v}\wedge \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \frac{\partial_{u} f }{\sqrt{u^{2} + v^{2}}} \boldsymbol{e}_{u} + \frac{\partial_{v} f }{\sqrt{u^{2} + v^{2}}} \boldsymbol{e}_{v} + \frac{\partial_{\phi } f }{u v} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot A = \frac{u A^{u} }{\left(u^{2} + v^{2}\right)^{\frac{3}{2}}} + \frac{v A^{v} }{\left(u^{2} + v^{2}\right)^{\frac{3}{2}}} + \frac{\partial_{u} A^{u} }{\sqrt{u^{2} + v^{2}}} + \frac{\partial_{v} A^{v} }{\sqrt{u^{2} + v^{2}}} + \frac{A^{v} }{v \sqrt{u^{2} + v^{2}}} + \frac{A^{u} }{u \sqrt{u^{2} + v^{2}}} + \frac{\partial_{\phi } A^{\phi } }{u v} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \W B = \left ( \frac{u B^{v\phi } }{\left(u^{2} + v^{2}\right)^{\frac{3}{2}}} - \frac{v B^{u\phi } }{\left(u^{2} + v^{2}\right)^{\frac{3}{2}}} - \frac{\partial_{v} B^{u\phi } }{\sqrt{u^{2} + v^{2}}} + \frac{\partial_{u} B^{v\phi } }{\sqrt{u^{2} + v^{2}}} - \frac{B^{u\phi } }{v \sqrt{u^{2} + v^{2}}} + \frac{B^{v\phi } }{u \sqrt{u^{2} + v^{2}}} + \frac{\partial_{\phi } B^{uv} }{u v}\right ) \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{v}\wedge \boldsymbol{e}_{\phi } \end{equation*}
    Derivatives in Prolate Spheroidal Coordinates
    \begin{equation*} f = f \end{equation*}
    \begin{equation*} A = A^{\xi }  \boldsymbol{e}_{\xi } + A^{\eta }  \boldsymbol{e}_{\eta } + A^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} B = B^{\xi \eta }  \boldsymbol{e}_{\xi }\wedge \boldsymbol{e}_{\eta } + B^{\xi \phi }  \boldsymbol{e}_{\xi }\wedge \boldsymbol{e}_{\phi } + B^{\eta \phi }  \boldsymbol{e}_{\eta }\wedge \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \frac{\partial_{\xi } f }{\sqrt{{\sin{\left (\eta  \right )}}^{2} + {\sinh{\left (\xi  \right )}}^{2}} \left|{a}\right|} \boldsymbol{e}_{\xi } + \frac{\partial_{\eta } f }{\sqrt{{\sin{\left (\eta  \right )}}^{2} + {\sinh{\left (\xi  \right )}}^{2}} \left|{a}\right|} \boldsymbol{e}_{\eta } + \frac{\partial_{\phi } f }{a \sin{\left (\eta  \right )} \sinh{\left (\xi  \right )}} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot A = \frac{a \left({\sin{\left (\eta  \right )}}^{2} + {\sinh{\left (\xi  \right )}}^{2}\right)^{3} \partial_{\phi } A^{\phi }  + \frac{\left(A^{\eta }  \sin{\left (2 \eta  \right )} + A^{\xi }  \sinh{\left (2 \xi  \right )}\right) \left({\sin{\left (\eta  \right )}}^{2} + {\sinh{\left (\xi  \right )}}^{2}\right)^{\frac{3}{2}} \sin{\left (\eta  \right )} \sinh{\left (\xi  \right )} \left|{a}\right|}{2} + \left({\sin{\left (\eta  \right )}}^{2} + {\sinh{\left (\xi  \right )}}^{2}\right)^{\frac{5}{2}} \left(\partial_{\eta } A^{\eta }  + \partial_{\xi } A^{\xi } \right) \sin{\left (\eta  \right )} \sinh{\left (\xi  \right )} \left|{a}\right| + \left({\sin{\left (\eta  \right )}}^{2} + {\sinh{\left (\xi  \right )}}^{2}\right)^{\frac{5}{2}} A^{\eta }  \cos{\left (\eta  \right )} \sinh{\left (\xi  \right )} \left|{a}\right| + \left({\sin{\left (\eta  \right )}}^{2} + {\sinh{\left (\xi  \right )}}^{2}\right)^{\frac{5}{2}} A^{\xi }  \sin{\left (\eta  \right )} \cosh{\left (\xi  \right )} \left|{a}\right|}{a^{2} \left({\sin{\left (\eta  \right )}}^{2} + {\sinh{\left (\xi  \right )}}^{2}\right)^{3} \sin{\left (\eta  \right )} \sinh{\left (\xi  \right )}} \end{equation*}
    \end{document}

```python
check('dchk')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} \left[\begin{array}{ccc}\left (\boldsymbol{e}_{a}\cdot \boldsymbol{e}_{a}\right )  & 0 & \left (\boldsymbol{e}_{a}\cdot \boldsymbol{e}_{b}\right ) \\0 & \left (\boldsymbol{e}_{ab}\cdot \boldsymbol{e}_{ab}\right )  & 0\\\left (\boldsymbol{e}_{a}\cdot \boldsymbol{e}_{b}\right )  & 0 & \left (\boldsymbol{e}_{b}\cdot \boldsymbol{e}_{b}\right ) \end{array}\right] \end{equation*}
    \begin{equation*} v^{a} \boldsymbol{e}_{a} + v^{ab} \boldsymbol{e}_{ab} + v^{b} \boldsymbol{e}_{b} \end{equation*}
    \begin{equation*} B^{aab} \boldsymbol{e}_{a}\wedge \boldsymbol{e}_{ab} + B^{ab} \boldsymbol{e}_{a}\wedge \boldsymbol{e}_{b} + B^{abb} \boldsymbol{e}_{ab}\wedge \boldsymbol{e}_{b} \end{equation*}
    \end{document}

```python
check('diffeq_sys')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def main():
        Print_Function()
        (a, b, c) = abc = symbols('a,b,c',real=True)
        (o3d, ea, eb, ec) = Ga.build('e_a e_b e_c', g=[1, 1, 1], coords=abc)
        grad = o3d.grad
        x = symbols('x',real=True)
        A = o3d.lt([[x*a*c**2,x**2*a*b*c,x**2*a**3*b**5],\
                    [x**3*a**2*b*c,x**4*a*b**2*c**5,5*x**4*a*b**2*c],\
                    [x**4*a*b**2*c**4,4*x**4*a*b**2*c**2,4*x**4*a**5*b**2*c]])
        print('A =',A)
        v = a*ea+b*eb+c*ec
        print('v =',v)
        f = v|A(v)
        print(r'%f = v\cdot \f{A}{v} =',f)
        (grad * f).Fmt(3,r'%\nabla f')
        Av = A(v)
        print(r'%\f{A}{v} =', Av)
        (grad * Av).Fmt(3,r'%\nabla \f{A}{v}')
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} A = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{a}\right ) =& a c^{2} x \boldsymbol{e}_{a} + a b c x^{2} \boldsymbol{e}_{b} + a^{3} b^{5} x^{2} \boldsymbol{e}_{c} \\ L \left ( \boldsymbol{e}_{b}\right ) =& a^{2} b c x^{3} \boldsymbol{e}_{a} + a b^{2} c^{5} x^{4} \boldsymbol{e}_{b} + 5 a b^{2} c x^{4} \boldsymbol{e}_{c} \\ L \left ( \boldsymbol{e}_{c}\right ) =& a b^{2} c^{4} x^{4} \boldsymbol{e}_{a} + 4 a b^{2} c^{2} x^{4} \boldsymbol{e}_{b} + 4 a^{5} b^{2} c x^{4} \boldsymbol{e}_{c}  \end{array} \right \} \end{equation*}
    \begin{equation*} v = a \boldsymbol{e}_{a} + b \boldsymbol{e}_{b} + c \boldsymbol{e}_{c} \end{equation*}
    \begin{equation*} f = v\cdot \f{A}{v} = a c x \left(4 a^{4} b^{2} c^{2} x^{3} + a^{3} b^{5} x + a^{2} b^{2} x^{2} + a^{2} c + a b^{2} c^{4} x^{3} + a b^{2} x + b^{4} c^{4} x^{3} + 4 b^{3} c^{2} x^{3} + 5 b^{3} c x^{3}\right) \end{equation*}
    \begin{equation*} \f{A}{v} = a c x \left(a b^{2} x^{2} + a c + b^{2} c^{4} x^{3}\right) \boldsymbol{e}_{a} + a b c x^{2} \left(a + b^{2} c^{4} x^{2} + 4 b c^{2} x^{2}\right) \boldsymbol{e}_{b} + a b^{2} x^{2} \left(4 a^{4} c^{2} x^{2} + a^{3} b^{3} + 5 b c x^{2}\right) \boldsymbol{e}_{c} \end{equation*}
    \end{document}

```python
check('dirac_derive')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} \psi   + \psi ^{01} \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{1} + \psi ^{02} \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{2} + \psi ^{03} \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{3} + \psi ^{12} \boldsymbol{\gamma }_{1}\wedge \boldsymbol{\gamma }_{2} + \psi ^{13} \boldsymbol{\gamma }_{1}\wedge \boldsymbol{\gamma }_{3} + \psi ^{23} \boldsymbol{\gamma }_{2}\wedge \boldsymbol{\gamma }_{3} + \psi ^{0123} \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{1}\wedge \boldsymbol{\gamma }_{2}\wedge \boldsymbol{\gamma }_{3} \end{equation*}
    \begin{equation*} B^{01} \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{1} + B^{02} \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{2} + B^{03} \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{3} + B^{12} \boldsymbol{\gamma }_{1}\wedge \boldsymbol{\gamma }_{2} + B^{13} \boldsymbol{\gamma }_{1}\wedge \boldsymbol{\gamma }_{3} + B^{23} \boldsymbol{\gamma }_{2}\wedge \boldsymbol{\gamma }_{3} \end{equation*}
    \begin{equation*} \left ( - 2 \left (\boldsymbol{\gamma }_{2}\cdot \boldsymbol{\gamma }_{2}\right )  B^{02} \psi ^{12} + 2 \left (\boldsymbol{\gamma }_{2}\cdot \boldsymbol{\gamma }_{2}\right )  B^{12} \psi ^{02} - 2 \left (\boldsymbol{\gamma }_{3}\cdot \boldsymbol{\gamma }_{3}\right )  B^{03} \psi ^{13} + 2 \left (\boldsymbol{\gamma }_{3}\cdot \boldsymbol{\gamma }_{3}\right )  B^{13} \psi ^{03}\right ) \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{1} + \left ( 2 \left (\boldsymbol{\gamma }_{1}\cdot \boldsymbol{\gamma }_{1}\right )  B^{01} \psi ^{12} - 2 \left (\boldsymbol{\gamma }_{1}\cdot \boldsymbol{\gamma }_{1}\right )  B^{12} \psi ^{01} - 2 \left (\boldsymbol{\gamma }_{3}\cdot \boldsymbol{\gamma }_{3}\right )  B^{03} \psi ^{23} + 2 \left (\boldsymbol{\gamma }_{3}\cdot \boldsymbol{\gamma }_{3}\right )  B^{23} \psi ^{03}\right ) \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{2} + \left ( 2 \left (\boldsymbol{\gamma }_{1}\cdot \boldsymbol{\gamma }_{1}\right )  B^{01} \psi ^{13} - 2 \left (\boldsymbol{\gamma }_{1}\cdot \boldsymbol{\gamma }_{1}\right )  B^{13} \psi ^{01} + 2 \left (\boldsymbol{\gamma }_{2}\cdot \boldsymbol{\gamma }_{2}\right )  B^{02} \psi ^{23} - 2 \left (\boldsymbol{\gamma }_{2}\cdot \boldsymbol{\gamma }_{2}\right )  B^{23} \psi ^{02}\right ) \boldsymbol{\gamma }_{0}\wedge \boldsymbol{\gamma }_{3} + \left ( - 2 \left (\boldsymbol{\gamma }_{0}\cdot \boldsymbol{\gamma }_{0}\right )  B^{01} \psi ^{02} + 2 \left (\boldsymbol{\gamma }_{0}\cdot \boldsymbol{\gamma }_{0}\right )  B^{02} \psi ^{01} - 2 \left (\boldsymbol{\gamma }_{3}\cdot \boldsymbol{\gamma }_{3}\right )  B^{13} \psi ^{23} + 2 \left (\boldsymbol{\gamma }_{3}\cdot \boldsymbol{\gamma }_{3}\right )  B^{23} \psi ^{13}\right ) \boldsymbol{\gamma }_{1}\wedge \boldsymbol{\gamma }_{2} + \left ( - 2 \left (\boldsymbol{\gamma }_{0}\cdot \boldsymbol{\gamma }_{0}\right )  B^{01} \psi ^{03} + 2 \left (\boldsymbol{\gamma }_{0}\cdot \boldsymbol{\gamma }_{0}\right )  B^{03} \psi ^{01} + 2 \left (\boldsymbol{\gamma }_{2}\cdot \boldsymbol{\gamma }_{2}\right )  B^{12} \psi ^{23} - 2 \left (\boldsymbol{\gamma }_{2}\cdot \boldsymbol{\gamma }_{2}\right )  B^{23} \psi ^{12}\right ) \boldsymbol{\gamma }_{1}\wedge \boldsymbol{\gamma }_{3} + \left ( - 2 \left (\boldsymbol{\gamma }_{0}\cdot \boldsymbol{\gamma }_{0}\right )  B^{02} \psi ^{03} + 2 \left (\boldsymbol{\gamma }_{0}\cdot \boldsymbol{\gamma }_{0}\right )  B^{03} \psi ^{02} - 2 \left (\boldsymbol{\gamma }_{1}\cdot \boldsymbol{\gamma }_{1}\right )  B^{12} \psi ^{13} + 2 \left (\boldsymbol{\gamma }_{1}\cdot \boldsymbol{\gamma }_{1}\right )  B^{13} \psi ^{12}\right ) \boldsymbol{\gamma }_{2}\wedge \boldsymbol{\gamma }_{3} \end{equation*}
    \end{document}

```python
check('Dop')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={6in,7in},total={5in,6in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} \nabla = \boldsymbol{e}_{x} \frac{\partial}{\partial x} + \boldsymbol{e}_{y} \frac{\partial}{\partial y} + \boldsymbol{e}_{z} \frac{\partial}{\partial z} \end{equation*}
    \begin{equation*} \nabla^{2} = \nabla \cdot \nabla = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}} \end{equation*}
    \begin{equation*} \lp\nabla^{2}\rp f = \partial^{2}_{x} f  + \partial^{2}_{y} f  + \partial^{2}_{z} f \end{equation*}
    \begin{equation*} \nabla\cdot\lp\nabla f\rp = \partial^{2}_{x} f  + \partial^{2}_{y} f  + \partial^{2}_{z} f \end{equation*}
    \begin{equation*} \nabla^{2} = \nabla\cdot\nabla = \frac{2}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2} \tan{\left (\theta  \right )}} \frac{\partial}{\partial \theta } + \frac{1}{r^{2} {\sin{\left (\theta  \right )}}^{2}} \frac{\partial^{2}}{\partial \phi ^{2}} + \frac{\partial^{2}}{\partial r^{2}} + r^{-2} \frac{\partial^{2}}{\partial \theta ^{2}} \end{equation*}
    \begin{equation*} \lp\nabla^{2}\rp f = \frac{r^{2} \partial^{2}_{r} f  + 2 r \partial_{r} f  + \partial^{2}_{\theta } f  + \frac{\partial_{\theta } f }{\tan{\left (\theta  \right )}} + \frac{\partial^{2}_{\phi } f }{{\sin{\left (\theta  \right )}}^{2}}}{r^{2}} \end{equation*}
    \begin{equation*} \nabla\cdot\lp\nabla f\rp = \frac{r^{2} \partial^{2}_{r} f  + 2 r \partial_{r} f  + \partial^{2}_{\theta } f  + \frac{\partial_{\theta } f }{\tan{\left (\theta  \right )}} + \frac{\partial^{2}_{\phi } f }{{\sin{\left (\theta  \right )}}^{2}}}{r^{2}} \end{equation*}
    \begin{equation*} \begin{array}{c} \left [ \boldsymbol{e}_{x} \frac{\partial}{\partial x} + \boldsymbol{e}_{y} \frac{\partial}{\partial y} + \boldsymbol{e}_{z} \frac{\partial}{\partial z}, \right. \\  \left. \boldsymbol{e}_{x} \frac{\partial}{\partial x} + \boldsymbol{e}_{y} \frac{\partial}{\partial y} + \boldsymbol{e}_{z} \frac{\partial}{\partial z}\right ] \\ \end{array} \end{equation*}
    \begin{equation*} F \end{equation*}
    \begin{equation*} F^{r}  \boldsymbol{e}_{r} + F^{\theta }  \boldsymbol{e}_{\theta } + F^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} F \end{equation*}
    \begin{equation*} F^{r}  \boldsymbol{e}_{r} + F^{\theta }  \boldsymbol{e}_{\theta } + F^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} F \end{equation*}
    \begin{equation*} \begin{array}{c} \left ( F^{r}  \boldsymbol{e}_{r} + F^{\theta }  \boldsymbol{e}_{\theta } + F^{\phi }  \boldsymbol{e}_{\phi }, \right. \\  \left. F^{r}  \boldsymbol{e}_{r} + F^{\theta }  \boldsymbol{e}_{\theta } + F^{\phi }  \boldsymbol{e}_{\phi }\right ) \\ \end{array} \end{equation*}
    \end{document}

```python
check('em_waves_latex')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} \text{Pseudo Scalar\;\;}I =  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} I_{xyz} =  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} \text{Electromagnetic Field Bi-Vector\;\;} F =  \begin{aligned}[t]  & - E^{x} e^{- i \left(k_{x} x + k_{y} y + k_{z} z - \omega  t\right)} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} \\  &  - E^{y} e^{- i \left(k_{x} x + k_{y} y + k_{z} z - \omega  t\right)} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y} \\  &  - E^{z} e^{- i \left(k_{x} x + k_{y} y + k_{z} z - \omega  t\right)} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} \\  &  - B^{z} e^{- i \left(k_{x} x + k_{y} y + k_{z} z - \omega  t\right)} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} \\  &  + B^{y} e^{i \left(- k_{x} x - k_{y} y - k_{z} z + \omega  t\right)} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} \\  &  - B^{x} e^{- i \left(k_{x} x + k_{y} y + k_{z} z - \omega  t\right)} \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}  \end{aligned} \end{equation*}
    Geom Derivative of Electomagnetic Field Bi-Vector
    \begin{equation*} \boldsymbol{\nabla}  F = 0 =  \begin{aligned}[t]  & - i \left(E^{x} k_{x} + E^{y} k_{y} + E^{z} k_{z}\right) e^{- i \left(k_{x} x + k_{y} y + k_{z} z - \omega  t\right)} \boldsymbol{\gamma }_{t} \\  &  + i \left(B^{y} k_{z} - B^{z} k_{y} - E^{x} \omega \right) e^{i \left(- k_{x} x - k_{y} y - k_{z} z + \omega  t\right)} \boldsymbol{\gamma }_{x} \\  &  + i \left(- B^{x} k_{z} + B^{z} k_{x} - E^{y} \omega \right) e^{i \left(- k_{x} x - k_{y} y - k_{z} z + \omega  t\right)} \boldsymbol{\gamma }_{y} \\  &  + i \left(B^{x} k_{y} - B^{y} k_{x} - E^{z} \omega \right) e^{i \left(- k_{x} x - k_{y} y - k_{z} z + \omega  t\right)} \boldsymbol{\gamma }_{z} \\  &  + i \left(- B^{z} \omega  - E^{x} k_{y} + E^{y} k_{x}\right) e^{i \left(- k_{x} x - k_{y} y - k_{z} z + \omega  t\right)} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} \\  &  + i \left(B^{y} \omega  - E^{x} k_{z} + E^{z} k_{x}\right) e^{i \left(- k_{x} x - k_{y} y - k_{z} z + \omega  t\right)} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} \\  &  + i \left(- B^{x} \omega  - E^{y} k_{z} + E^{z} k_{y}\right) e^{i \left(- k_{x} x - k_{y} y - k_{z} z + \omega  t\right)} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \\  &  - i \left(B^{x} k_{x} + B^{y} k_{y} + B^{z} k_{z}\right) e^{- i \left(k_{x} x + k_{y} y + k_{z} z - \omega  t\right)} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}  \end{aligned} \end{equation*}
    \begin{equation*} \lp\bm{\nabla}F\rp /\lp i e^{iK\cdot X}\rp = 0 =  \begin{aligned}[t]  & \left ( - E^{x} k_{x} - E^{y} k_{y} - E^{z} k_{z}\right ) \boldsymbol{\gamma }_{t} \\  &  + \left ( B^{y} k_{z} - B^{z} k_{y} - E^{x} \omega \right ) \boldsymbol{\gamma }_{x} \\  &  + \left ( - B^{x} k_{z} + B^{z} k_{x} - E^{y} \omega \right ) \boldsymbol{\gamma }_{y} \\  &  + \left ( B^{x} k_{y} - B^{y} k_{x} - E^{z} \omega \right ) \boldsymbol{\gamma }_{z} \\  &  + \left ( - B^{z} \omega  - E^{x} k_{y} + E^{y} k_{x}\right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} \\  &  + \left ( B^{y} \omega  - E^{x} k_{z} + E^{z} k_{x}\right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} \\  &  + \left ( - B^{x} \omega  - E^{y} k_{z} + E^{z} k_{y}\right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \\  &  + \left ( - B^{x} k_{x} - B^{y} k_{y} - B^{z} k_{z}\right ) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}  \end{aligned} \end{equation*}
    \begin{equation*} \mbox{set } e_{E}\cdot e_{k} = e_{B}\cdot e_{k} = 0\mbox{ and } e_{E}\cdot e_{E} = e_{B}\cdot e_{B} = e_{k}\cdot e_{k} = -e_{t}\cdot e_{t} = 1 \end{equation*}
    \begin{equation*} g = \left[\begin{array}{cccc}-1 & \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  & 0 & 0\\\left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & 1\end{array}\right] \end{equation*}
    \begin{equation*} K\cdot X = - k x_{k} + \omega  t \end{equation*}
    \begin{equation*} F = - \frac{B e^{- i \left(k x_{k} - \omega  t\right)}}{\sqrt{1 - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right ) ^{2}}} \boldsymbol{e}_{E}\wedge \boldsymbol{e}_{k} + E e^{i \left(- k x_{k} + \omega  t\right)} \boldsymbol{e}_{E}\wedge \boldsymbol{t} - \frac{\left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  B e^{- i \left(k x_{k} - \omega  t\right)}}{\sqrt{1 - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right ) ^{2}}} \boldsymbol{e}_{B}\wedge \boldsymbol{e}_{k} \end{equation*}
    \begin{equation*} \lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0 =  \begin{aligned}[t]  & \left ( - \frac{B k}{\sqrt{1 - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right ) ^{2}}} - E \omega \right ) \boldsymbol{e}_{E} \\  &  - \frac{\left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  B k}{\sqrt{1 - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right ) ^{2}}} \boldsymbol{e}_{B} \\  &  + \left ( - \frac{B \omega }{\sqrt{1 - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right ) ^{2}}} - E k\right ) \boldsymbol{e}_{E}\wedge \boldsymbol{e}_{k}\wedge \boldsymbol{t} \\  &  - \frac{\left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  B \omega }{\sqrt{1 - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right ) ^{2}}} \boldsymbol{e}_{B}\wedge \boldsymbol{e}_{k}\wedge \boldsymbol{t}  \end{aligned} \end{equation*}
    \begin{equation*} \mbox{Previous equation requires that: }e_{E}\cdot e_{B} = 0\mbox{ if }B\ne 0\mbox{ and }k\ne 0 \end{equation*}
    \begin{equation*} \lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X} \rp = 0 =  \begin{aligned}[t]  & \left ( - B k - E \omega \right ) \boldsymbol{e}_{E} \\  &  + \left ( - B \omega  - E k\right ) \boldsymbol{e}_{E}\wedge \boldsymbol{e}_{k}\wedge \boldsymbol{t}  \end{aligned} \end{equation*}
    \begin{equation*} 0 = - B k - E \omega \end{equation*}
    \begin{equation*} 0 = - B \omega  - E k \end{equation*}
    \begin{equation*} \mbox{eq3 = eq1-eq2: }0 = \frac{E k}{\omega } - \frac{E \omega }{k} \end{equation*}
    \begin{equation*} \mbox{eq3 = (eq1-eq2)/E: }0 = \frac{k}{\omega } - \frac{\omega }{k} \end{equation*}
    \begin{equation*} k = \left[\begin{array}{c}- \omega \\\omega \end{array}\right] \end{equation*}
    \begin{equation*} B = \left[\begin{array}{c}- E\\E\end{array}\right] \end{equation*}
    \end{document}

```python
check('FmtChk')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={6in,7in},total={5in,6in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} \begin{array}{c} \left [ f , \right. \\  F^{x}  \boldsymbol{e}_{x} + F^{y}  \boldsymbol{e}_{y} + F^{z}  \boldsymbol{e}_{z}, \\ \left. B^{xy}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + B^{xz}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + B^{yz}  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}\right ] \\ \end{array} \end{equation*}
    \begin{equation*} \left [ \begin{array}{ccc} f , & F^{x}  \boldsymbol{e}_{x} + F^{y}  \boldsymbol{e}_{y} + F^{z}  \boldsymbol{e}_{z}, & B^{xy}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + B^{xz}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + B^{yz}  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}\\ \end{array} \right ] \end{equation*}
    \begin{equation*} \begin{aligned}[t]  & F^{x}  \boldsymbol{e}_{x} \\  &  + F^{y}  \boldsymbol{e}_{y} \\  &  + F^{z}  \boldsymbol{e}_{z}  \end{aligned} \end{equation*}
    \begin{equation*} \begin{aligned}[t]  & B^{xy}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \\  &  + B^{xz}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} \\  &  + B^{yz}  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}  \end{aligned} \end{equation*}
    \begin{equation*} \nabla^{2} = \nabla\cdot\nabla = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}} \end{equation*}
    \begin{equation*} \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}} + \boldsymbol{e}_{x} \frac{\partial}{\partial x} + \boldsymbol{e}_{y} \frac{\partial}{\partial y} + \boldsymbol{e}_{z} \frac{\partial}{\partial z} \end{equation*}
    \end{document}

```python
check('groups')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={8.5in,11in},total={7.5in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def Product_of_Rotors():
        Print_Function()
        (na,nb,nm,alpha,th,th_a,th_b) = symbols('n_a n_b n_m alpha theta theta_a theta_b',\
                                        real = True)
        g = [[na, 0, alpha],[0, nm, 0],[alpha, 0, nb]] #metric tensor
        """
        Values of metric tensor components
        [na,nm,nb] = [+1/-1,+1/-1,+1/-1]  alpha = ea|eb
        """
        (g3d, ea, em, eb) = Ga.build('e_a e_m e_b', g=g)
        print('g =',g3d.g)
        print(r'%n_{a} = \bm{e}_{a}^{2}\;\;n_{b} = \bm{e}_{b}^{2}\;\;n_{m} = \bm{e}_{m}^{2}'+\
            r'\;\;\alpha = \bm{e}_{a}\cdot\bm{e}_{b}')
        (ca,cb,sa,sb) = symbols('c_a c_b s_a s_b',real=True)
        Ra = ca + sa*ea*em  # Rotor for ea^em plane
        Rb = cb + sb*em*eb  # Rotor for em^eb plane
        print(r'%\mbox{Rotor in }\bm{e}_{a}\bm{e}_{m}\mbox{ plane } R_{a} =',Ra)
        print(r'%\mbox{Rotor in }\bm{e}_{m}\bm{e}_{b}\mbox{ plane } R_{b} =',Rb)
        Rab = Ra*Rb  # Compound Rotor
        """
        Show that compound rotor is scalar plus bivector
        """
        print(r'%R_{a}R_{b} = S+\bm{B} =', Rab)
        Rab2 = Rab.get_grade(2)
        print(r'%\bm{B} =',Rab2)
        Rab2sq = Rab2*Rab2  # Square of compound rotor bivector part
        Ssq = (Rab.scalar())**2  # Square of compound rotor scalar part
        Bsq =  Rab2sq.scalar()
        print(r'%S^{2} =',Ssq)
        print(r'%\bm{B}^{2} =',Bsq)
        Dsq = (Ssq-Bsq).expand().simplify()
        print('%S^{2}-B^{2} =', Dsq)
        Dsq = Dsq.subs(nm**2,S(1))  # (e_m)**4 = 1
        print('%S^{2}-B^{2} =', Dsq)
        Cases = [S(-1),S(1)]  # -1/+1 squares for each basis vector
        print(r'#Consider all combinations of $\bm{e}_{a}^{2}$, $\bm{e}_{b}^{2}$'+\
              r' and $\bm{e}_{m}^2$:')
        for Na in Cases:
            for Nb in Cases:
                for Nm in Cases:
                    Ba_sq = -Na*Nm
                    Bb_sq = -Nb*Nm
                    if Ba_sq < 0:
                        Ca_th = cos(th_a)
                        Sa_th = sin(th_a)
                    else:
                        Ca_th = cosh(th_a)
                        Sa_th = sinh(th_a)
                    if Bb_sq < 0:
                        Cb_th = cos(th_b)
                        Sb_th = sin(th_b)
                    else:
                        Cb_th = cosh(th_b)
                        Sb_th = sinh(th_b)
                    print(r'%\left [ \bm{e}_{a}^{2},\bm{e}_{b}^{2},\bm{e}_{m}^2\right ] =',\
                          [Na,Nb,Nm])
                    Dsq_tmp = Dsq.subs({ca:Ca_th,sa:Sa_th,cb:Cb_th,sb:Sb_th,na:Na,nb:Nb,nm:Nm})
                    print(r'%S^{2}-\bm{B}^{2} =',Dsq_tmp,' =',trigsimp(Dsq_tmp))
        print(r'#Thus we have shown that $R_{a}R_{b} = S+\bm{D} = e^{\bm{C}}$ where $\bm{C}$'+\
              r' is a bivector blade.')
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} g = \left[\begin{array}{ccc}n_{a} & 0 & \alpha \\0 & n_{m} & 0\\\alpha  & 0 & n_{b}\end{array}\right] \end{equation*}
    \begin{equation*} n_{a} = \bm{e}_{a}^{2}\;\;n_{b} = \bm{e}_{b}^{2}\;\;n_{m} = \bm{e}_{m}^{2}\;\;\alpha = \bm{e}_{a}\cdot\bm{e}_{b} \end{equation*}
    \begin{equation*} \mbox{Rotor in }\bm{e}_{a}\bm{e}_{m}\mbox{ plane } R_{a} = c_{a}  + s_{a} \boldsymbol{e}_{a}\wedge \boldsymbol{e}_{m} \end{equation*}
    \begin{equation*} \mbox{Rotor in }\bm{e}_{m}\bm{e}_{b}\mbox{ plane } R_{b} = c_{b}  + s_{b} \boldsymbol{e}_{m}\wedge \boldsymbol{e}_{b} \end{equation*}
    \begin{equation*} R_{a}R_{b} = S+\bm{B} = \left ( \alpha  n_{m} s_{a} s_{b} + c_{a} c_{b}\right )  + c_{b} s_{a} \boldsymbol{e}_{a}\wedge \boldsymbol{e}_{m} + n_{m} s_{a} s_{b} \boldsymbol{e}_{a}\wedge \boldsymbol{e}_{b} + c_{a} s_{b} \boldsymbol{e}_{m}\wedge \boldsymbol{e}_{b} \end{equation*}
    \begin{equation*} \bm{B} = c_{b} s_{a} \boldsymbol{e}_{a}\wedge \boldsymbol{e}_{m} + n_{m} s_{a} s_{b} \boldsymbol{e}_{a}\wedge \boldsymbol{e}_{b} + c_{a} s_{b} \boldsymbol{e}_{m}\wedge \boldsymbol{e}_{b} \end{equation*}
    \begin{equation*} S^{2} = \left(\alpha  n_{m} s_{a} s_{b} + c_{a} c_{b}\right)^{2} \end{equation*}
    \begin{equation*} \bm{B}^{2} = \alpha ^{2} {\left ( n_{m} \right )}^{2} {\left ( s_{a} \right )}^{2} {\left ( s_{b} \right )}^{2} + 2 \alpha  c_{a} c_{b} n_{m} s_{a} s_{b} - {\left ( c_{a} \right )}^{2} n_{b} n_{m} {\left ( s_{b} \right )}^{2} - {\left ( c_{b} \right )}^{2} n_{a} n_{m} {\left ( s_{a} \right )}^{2} - n_{a} n_{b} {\left ( n_{m} \right )}^{2} {\left ( s_{a} \right )}^{2} {\left ( s_{b} \right )}^{2} \end{equation*}
    \begin{equation*} S^{2}-B^{2} = {\left ( c_{a} \right )}^{2} {\left ( c_{b} \right )}^{2} + {\left ( c_{a} \right )}^{2} n_{b} n_{m} {\left ( s_{b} \right )}^{2} + {\left ( c_{b} \right )}^{2} n_{a} n_{m} {\left ( s_{a} \right )}^{2} + n_{a} n_{b} {\left ( n_{m} \right )}^{2} {\left ( s_{a} \right )}^{2} {\left ( s_{b} \right )}^{2} \end{equation*}
    \begin{equation*} S^{2}-B^{2} = {\left ( c_{a} \right )}^{2} {\left ( c_{b} \right )}^{2} + {\left ( c_{a} \right )}^{2} n_{b} n_{m} {\left ( s_{b} \right )}^{2} + {\left ( c_{b} \right )}^{2} n_{a} n_{m} {\left ( s_{a} \right )}^{2} + n_{a} n_{b} {\left ( s_{a} \right )}^{2} {\left ( s_{b} \right )}^{2} \end{equation*}
    Consider all combinations of $\bm{e}_{a}^{2}$, $\bm{e}_{b}^{2}$ and $\bm{e}_{m}^2$:
    \begin{equation*} \left [ \bm{e}_{a}^{2},\bm{e}_{b}^{2},\bm{e}_{m}^2\right ] = [-1, -1, -1] \end{equation*}
    \begin{equation*} S^{2}-\bm{B}^{2} = {\sin{\left (\theta _{a} \right )}}^{2} {\sin{\left (\theta _{b} \right )}}^{2} + {\sin{\left (\theta _{a} \right )}}^{2} {\cos{\left (\theta _{b} \right )}}^{2} + {\sin{\left (\theta _{b} \right )}}^{2} {\cos{\left (\theta _{a} \right )}}^{2} + {\cos{\left (\theta _{a} \right )}}^{2} {\cos{\left (\theta _{b} \right )}}^{2}  = 1 \end{equation*}
    \begin{equation*} \left [ \bm{e}_{a}^{2},\bm{e}_{b}^{2},\bm{e}_{m}^2\right ] = [-1, -1, 1] \end{equation*}
    \begin{equation*} S^{2}-\bm{B}^{2} = {\sinh{\left (\theta _{a} \right )}}^{2} {\sinh{\left (\theta _{b} \right )}}^{2} - {\sinh{\left (\theta _{a} \right )}}^{2} {\cosh{\left (\theta _{b} \right )}}^{2} - {\sinh{\left (\theta _{b} \right )}}^{2} {\cosh{\left (\theta _{a} \right )}}^{2} + {\cosh{\left (\theta _{a} \right )}}^{2} {\cosh{\left (\theta _{b} \right )}}^{2}  = 1 \end{equation*}
    \begin{equation*} \left [ \bm{e}_{a}^{2},\bm{e}_{b}^{2},\bm{e}_{m}^2\right ] = [-1, 1, -1] \end{equation*}
    \begin{equation*} S^{2}-\bm{B}^{2} = - {\sin{\left (\theta _{a} \right )}}^{2} {\sinh{\left (\theta _{b} \right )}}^{2} + {\sin{\left (\theta _{a} \right )}}^{2} {\cosh{\left (\theta _{b} \right )}}^{2} - {\cos{\left (\theta _{a} \right )}}^{2} {\sinh{\left (\theta _{b} \right )}}^{2} + {\cos{\left (\theta _{a} \right )}}^{2} {\cosh{\left (\theta _{b} \right )}}^{2}  = 1 \end{equation*}
    \begin{equation*} \left [ \bm{e}_{a}^{2},\bm{e}_{b}^{2},\bm{e}_{m}^2\right ] = [-1, 1, 1] \end{equation*}
    \begin{equation*} S^{2}-\bm{B}^{2} = - {\sin{\left (\theta _{b} \right )}}^{2} {\sinh{\left (\theta _{a} \right )}}^{2} + {\sin{\left (\theta _{b} \right )}}^{2} {\cosh{\left (\theta _{a} \right )}}^{2} - {\cos{\left (\theta _{b} \right )}}^{2} {\sinh{\left (\theta _{a} \right )}}^{2} + {\cos{\left (\theta _{b} \right )}}^{2} {\cosh{\left (\theta _{a} \right )}}^{2}  = 1 \end{equation*}
    \begin{equation*} \left [ \bm{e}_{a}^{2},\bm{e}_{b}^{2},\bm{e}_{m}^2\right ] = [1, -1, -1] \end{equation*}
    \begin{equation*} S^{2}-\bm{B}^{2} = - {\sin{\left (\theta _{b} \right )}}^{2} {\sinh{\left (\theta _{a} \right )}}^{2} + {\sin{\left (\theta _{b} \right )}}^{2} {\cosh{\left (\theta _{a} \right )}}^{2} - {\cos{\left (\theta _{b} \right )}}^{2} {\sinh{\left (\theta _{a} \right )}}^{2} + {\cos{\left (\theta _{b} \right )}}^{2} {\cosh{\left (\theta _{a} \right )}}^{2}  = 1 \end{equation*}
    \begin{equation*} \left [ \bm{e}_{a}^{2},\bm{e}_{b}^{2},\bm{e}_{m}^2\right ] = [1, -1, 1] \end{equation*}
    \begin{equation*} S^{2}-\bm{B}^{2} = - {\sin{\left (\theta _{a} \right )}}^{2} {\sinh{\left (\theta _{b} \right )}}^{2} + {\sin{\left (\theta _{a} \right )}}^{2} {\cosh{\left (\theta _{b} \right )}}^{2} - {\cos{\left (\theta _{a} \right )}}^{2} {\sinh{\left (\theta _{b} \right )}}^{2} + {\cos{\left (\theta _{a} \right )}}^{2} {\cosh{\left (\theta _{b} \right )}}^{2}  = 1 \end{equation*}
    \begin{equation*} \left [ \bm{e}_{a}^{2},\bm{e}_{b}^{2},\bm{e}_{m}^2\right ] = [1, 1, -1] \end{equation*}
    \begin{equation*} S^{2}-\bm{B}^{2} = {\sinh{\left (\theta _{a} \right )}}^{2} {\sinh{\left (\theta _{b} \right )}}^{2} - {\sinh{\left (\theta _{a} \right )}}^{2} {\cosh{\left (\theta _{b} \right )}}^{2} - {\sinh{\left (\theta _{b} \right )}}^{2} {\cosh{\left (\theta _{a} \right )}}^{2} + {\cosh{\left (\theta _{a} \right )}}^{2} {\cosh{\left (\theta _{b} \right )}}^{2}  = 1 \end{equation*}
    \begin{equation*} \left [ \bm{e}_{a}^{2},\bm{e}_{b}^{2},\bm{e}_{m}^2\right ] = [1, 1, 1] \end{equation*}
    \begin{equation*} S^{2}-\bm{B}^{2} = {\sin{\left (\theta _{a} \right )}}^{2} {\sin{\left (\theta _{b} \right )}}^{2} + {\sin{\left (\theta _{a} \right )}}^{2} {\cos{\left (\theta _{b} \right )}}^{2} + {\sin{\left (\theta _{b} \right )}}^{2} {\cos{\left (\theta _{a} \right )}}^{2} + {\cos{\left (\theta _{a} \right )}}^{2} {\cos{\left (\theta _{b} \right )}}^{2}  = 1 \end{equation*}
    Thus we have shown that $R_{a}R_{b} = S+\bm{D} = e^{\bm{C}}$ where $\bm{C}$ is a bivector blade.
    \end{document}

```python
check('latex_check')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def basic_multivector_operations_3D():
        Print_Function()
        g3d = Ga('e*x|y|z')
        (ex,ey,ez) = g3d.mv()
        A = g3d.mv('A','mv')
        print(A.Fmt(1,'A'))
        print(A.Fmt(2,'A'))
        print(A.Fmt(3,'A'))
        print(A.even().Fmt(1,'%A_{+}'))
        print(A.odd().Fmt(1,'%A_{-}'))
        X = g3d.mv('X','vector')
        Y = g3d.mv('Y','vector')
        print('g_{ij} = ',g3d.g)
        print(X.Fmt(1,'X'))
        print(Y.Fmt(1,'Y'))
        print((X*Y).Fmt(2,'X*Y'))
        print((X^Y).Fmt(2,'X^Y'))
        print((X|Y).Fmt(2,'X|Y'))
        print(cross(X,Y).Fmt(1,r'X\times Y'))
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} A = A  + A^{x} \boldsymbol{e}_{x} + A^{y} \boldsymbol{e}_{y} + A^{z} \boldsymbol{e}_{z} + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + A^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + A^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} + A^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} A =  \begin{aligned}[t]  & A  \\  &  + A^{x} \boldsymbol{e}_{x} + A^{y} \boldsymbol{e}_{y} + A^{z} \boldsymbol{e}_{z} \\  &  + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + A^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + A^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \\  &  + A^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}  \end{aligned} \end{equation*}
    \begin{equation*} A =  \begin{aligned}[t]  & A  \\  &  + A^{x} \boldsymbol{e}_{x} \\  &  + A^{y} \boldsymbol{e}_{y} \\  &  + A^{z} \boldsymbol{e}_{z} \\  &  + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \\  &  + A^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} \\  &  + A^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \\  &  + A^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}  \end{aligned} \end{equation*}
    \begin{equation*} A_{+} = A  + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + A^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + A^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} A_{-} = A^{x} \boldsymbol{e}_{x} + A^{y} \boldsymbol{e}_{y} + A^{z} \boldsymbol{e}_{z} + A^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} g_{ij} =  \left[\begin{array}{ccc}\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  & \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  & \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right ) \\\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  & \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  & \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right ) \\\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  & \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  & \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right ) \end{array}\right] \end{equation*}
    \begin{equation*} X = X^{x} \boldsymbol{e}_{x} + X^{y} \boldsymbol{e}_{y} + X^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} Y = Y^{x} \boldsymbol{e}_{x} + Y^{y} \boldsymbol{e}_{y} + Y^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} X Y =  \begin{aligned}[t]  & \left ( \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} Y^{y} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{z} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{y} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{z} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{y} + \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{z}\right )  \\  &  + \left ( X^{x} Y^{y} - X^{y} Y^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( X^{x} Y^{z} - X^{z} Y^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( X^{y} Y^{z} - X^{z} Y^{y}\right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}  \end{aligned} \end{equation*}
    \begin{equation*} X\W Y = \left ( X^{x} Y^{y} - X^{y} Y^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( X^{x} Y^{z} - X^{z} Y^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( X^{y} Y^{z} - X^{z} Y^{y}\right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} X\cdot Y = \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} Y^{y} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{z} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{y} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{z} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{y} + \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{z} \end{equation*}
    \begin{equation*} X\times Y = \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{y} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{z} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{x} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{x} Y^{y} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{x} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{z} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{z} - \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{y} - \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right ) ^{2} X^{y} Y^{z} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right ) ^{2} X^{z} Y^{y}}{\sqrt{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right ) ^{2} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2} \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  + 2 \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right ) ^{2} \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right ) }} \boldsymbol{e}_{x} + \frac{- \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{y} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{x} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{z} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{y} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{x} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{z} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{y} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right ) ^{2} X^{x} Y^{z} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right ) ^{2} X^{z} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{z} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{y}}{\sqrt{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right ) ^{2} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2} \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  + 2 \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right ) ^{2} \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right ) }} \boldsymbol{e}_{y} + \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{x} Y^{y} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{z} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{x} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2} X^{x} Y^{y} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2} X^{y} Y^{x} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{z} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{z} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{y} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{z} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{z} Y^{y}}{\sqrt{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right ) ^{2} - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2} \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  + 2 \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right ) ^{2} \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right ) }} \boldsymbol{e}_{z} \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def basic_multivector_operations_2D():
        Print_Function()
        g2d = Ga('e*x|y')
        (ex,ey) = g2d.mv()
        print('g_{ij} =',g2d.g)
        X = g2d.mv('X','vector')
        A = g2d.mv('A','spinor')
        print(X.Fmt(1,'X'))
        print(A.Fmt(1,'A'))
        print((X|A).Fmt(2,'X|A'))
        print((X<A).Fmt(2,'X<A'))
        print((A>X).Fmt(2,'A>X'))
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} g_{ij} = \left[\begin{array}{cc}\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  & \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) \\\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  & \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right ) \end{array}\right] \end{equation*}
    \begin{equation*} X = X^{x} \boldsymbol{e}_{x} + X^{y} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} A = A  + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} X\cdot A = - A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{x} + A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} X\rfloor A = - A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{x} + A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} A\lfloor X = A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{x} - A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{y} \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def basic_multivector_operations_2D_orthogonal():
        Print_Function()
        o2d = Ga('e*x|y',g=[1,1])
        (ex,ey) = o2d.mv()
        print('g_{ii} =',o2d.g)
        X = o2d.mv('X','vector')
        A = o2d.mv('A','spinor')
        print(X.Fmt(1,'X'))
        print(A.Fmt(1,'A'))
        print((X*A).Fmt(2,'X*A'))
        print((X|A).Fmt(2,'X|A'))
        print((X<A).Fmt(2,'X<A'))
        print((X>A).Fmt(2,'X>A'))
        print((A*X).Fmt(2,'A*X'))
        print((A|X).Fmt(2,'A|X'))
        print((A<X).Fmt(2,'A<X'))
        print((A>X).Fmt(2,'A>X'))
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} g_{ii} = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right] \end{equation*}
    \begin{equation*} X = X^{x} \boldsymbol{e}_{x} + X^{y} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} A = A  + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} X A = \left ( A X^{x} - A^{xy} X^{y}\right ) \boldsymbol{e}_{x} + \left ( A X^{y} + A^{xy} X^{x}\right ) \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} X\cdot A = - A^{xy} X^{y} \boldsymbol{e}_{x} + A^{xy} X^{x} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} X\rfloor A = - A^{xy} X^{y} \boldsymbol{e}_{x} + A^{xy} X^{x} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} X\lfloor A = A X^{x} \boldsymbol{e}_{x} + A X^{y} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} A X = \left ( A X^{x} + A^{xy} X^{y}\right ) \boldsymbol{e}_{x} + \left ( A X^{y} - A^{xy} X^{x}\right ) \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} A\cdot X = A^{xy} X^{y} \boldsymbol{e}_{x} - A^{xy} X^{x} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} A\rfloor X = A X^{x} \boldsymbol{e}_{x} + A X^{y} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} A\lfloor X = A^{xy} X^{y} \boldsymbol{e}_{x} - A^{xy} X^{x} \boldsymbol{e}_{y} \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def check_generalized_BAC_CAB_formulas():
        Print_Function()
        g4d = Ga('a b c d e')
        (a,b,c,d,e) = g4d.mv()
        print('g_{ij} =',g4d.g)
        print('\\bm{a|(b*c)} =',a|(b*c))
        print('\\bm{a|(b^c)} =',a|(b^c))
        print('\\bm{a|(b^c^d)} =',a|(b^c^d))
        print('\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',(a|(b^c))+(c|(a^b))+(b|(c^a)))
        print('\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',a*(b^c)-b*(a^c)+c*(a^b))
        print('\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c))
        print('\\bm{(a^b)|(c^d)} =',(a^b)|(c^d))
        print('\\bm{((a^b)|c)|d} =',((a^b)|c)|d)
        print('\\bm{(a^b)\\times (c^d)} =',Ga.com(a^b,c^d))
        print('\\bm{(a^b^c)(d^e)} =',((a^b^c)*(d^e)).Fmt(2))
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} g_{ij} = \left[\begin{array}{ccccc}\left (\boldsymbol{a}\cdot \boldsymbol{a}\right )  & \left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  & \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{a}\cdot \boldsymbol{e}\right ) \\\left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{b}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{e}\right ) \\\left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{c}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{c}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{c}\cdot \boldsymbol{e}\right ) \\\left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{c}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{d}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{d}\cdot \boldsymbol{e}\right ) \\\left (\boldsymbol{a}\cdot \boldsymbol{e}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{e}\right )  & \left (\boldsymbol{c}\cdot \boldsymbol{e}\right )  & \left (\boldsymbol{d}\cdot \boldsymbol{e}\right )  & \left (\boldsymbol{e}\cdot \boldsymbol{e}\right ) \end{array}\right] \end{equation*}
    \begin{equation*} \bm{a\cdot (b c)} = - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \boldsymbol{b} + \left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  \boldsymbol{c} \end{equation*}
    \begin{equation*} \bm{a\cdot (b\W c)} = - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \boldsymbol{b} + \left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  \boldsymbol{c} \end{equation*}
    \begin{equation*} \bm{a\cdot (b\W c\W d)} = \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \boldsymbol{b}\wedge \boldsymbol{c} - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \boldsymbol{b}\wedge \boldsymbol{d} + \left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  \boldsymbol{c}\wedge \boldsymbol{d} \end{equation*}
    \begin{equation*} \bm{a\cdot (b\W c)+c\cdot (a\W b)+b\cdot (c\W a)} =  0 \end{equation*}
    \begin{equation*} \bm{a (b\W c)-b (a\W c)+c (a\W b)} = 3 \boldsymbol{a}\wedge \boldsymbol{b}\wedge \boldsymbol{c} \end{equation*}
    \begin{equation*} \bm{a (b\W c\W d)-b (a\W c\W d)+c (a\W b\W d)-d (a\W b\W c)} = 4 \boldsymbol{a}\wedge \boldsymbol{b}\wedge \boldsymbol{c}\wedge \boldsymbol{d} \end{equation*}
    \begin{equation*} \bm{(a\W b)\cdot (c\W d)} = - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  + \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{c}\right ) \end{equation*}
    \begin{equation*} \bm{((a\W b)\cdot c)\cdot d} = - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  + \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{c}\right ) \end{equation*}
    \begin{equation*} \bm{(a\W b)\times (c\W d)} = - \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  \boldsymbol{a}\wedge \boldsymbol{c} + \left (\boldsymbol{b}\cdot \boldsymbol{c}\right )  \boldsymbol{a}\wedge \boldsymbol{d} + \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \boldsymbol{b}\wedge \boldsymbol{c} - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \boldsymbol{b}\wedge \boldsymbol{d} \end{equation*}
    \begin{equation*} \bm{(a\W b\W c)(d\W e)} =  \begin{aligned}[t]  & \left ( - \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  \left (\boldsymbol{c}\cdot \boldsymbol{e}\right )  + \left (\boldsymbol{b}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{c}\cdot \boldsymbol{d}\right ) \right ) \boldsymbol{a} + \left ( \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \left (\boldsymbol{c}\cdot \boldsymbol{e}\right )  - \left (\boldsymbol{a}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{c}\cdot \boldsymbol{d}\right ) \right ) \boldsymbol{b} + \left ( - \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{e}\right )  + \left (\boldsymbol{a}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{d}\right ) \right ) \boldsymbol{c} \\  &  - \left (\boldsymbol{c}\cdot \boldsymbol{e}\right )  \boldsymbol{a}\wedge \boldsymbol{b}\wedge \boldsymbol{d} + \left (\boldsymbol{c}\cdot \boldsymbol{d}\right )  \boldsymbol{a}\wedge \boldsymbol{b}\wedge \boldsymbol{e} + \left (\boldsymbol{b}\cdot \boldsymbol{e}\right )  \boldsymbol{a}\wedge \boldsymbol{c}\wedge \boldsymbol{d} - \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  \boldsymbol{a}\wedge \boldsymbol{c}\wedge \boldsymbol{e} - \left (\boldsymbol{a}\cdot \boldsymbol{e}\right )  \boldsymbol{b}\wedge \boldsymbol{c}\wedge \boldsymbol{d} + \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \boldsymbol{b}\wedge \boldsymbol{c}\wedge \boldsymbol{e} \\  &  + \boldsymbol{a}\wedge \boldsymbol{b}\wedge \boldsymbol{c}\wedge \boldsymbol{d}\wedge \boldsymbol{e}  \end{aligned} \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def rounding_numerical_components():
        Print_Function()
        o3d = Ga('e_x e_y e_z',g=[1,1,1])
        (ex,ey,ez) = o3d.mv()
        X = 1.2*ex+2.34*ey+0.555*ez
        Y = 0.333*ex+4*ey+5.3*ez
        print('X =',X)
        print('Nga(X,2) =',Nga(X,2))
        print('X*Y =',X*Y)
        print('Nga(X*Y,2) =',Nga(X*Y,2))
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} X = 1.2 \boldsymbol{e}_{x} + 2.34 \boldsymbol{e}_{y} + 0.555 \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} Nga(X,2) = 1.2 \boldsymbol{e}_{x} + 2.3 \boldsymbol{e}_{y} + 0.55 \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} X Y = 12.7011  + 4.02078 \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + 6.175185 \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + 10.182 \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} Nga(X Y,2) = 13.0  + 4.0 \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + 6.2 \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + 10.0 \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def derivatives_in_rectangular_coordinates():
        Print_Function()
        X = (x,y,z) = symbols('x y z')
        o3d = Ga('e_x e_y e_z',g=[1,1,1],coords=X)
        (ex,ey,ez) = o3d.mv()
        grad = o3d.grad
        f = o3d.mv('f','scalar',f=True)
        A = o3d.mv('A','vector',f=True)
        B = o3d.mv('B','bivector',f=True)
        C = o3d.mv('C','mv')
        print('f =',f)
        print('A =',A)
        print('B =',B)
        print('C =',C)
        print('grad*f =',grad*f)
        print('grad|A =',grad|A)
        print('grad*A =',grad*A)
        print('-I*(grad^A) =',-o3d.I()*(grad^A))
        print('grad*B =',grad*B)
        print('grad^B =',grad^B)
        print('grad|B =',grad|B)
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} f = f \end{equation*}
    \begin{equation*} A = A^{x}  \boldsymbol{e}_{x} + A^{y}  \boldsymbol{e}_{y} + A^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} B = B^{xy}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + B^{xz}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + B^{yz}  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} C = C  + C^{x} \boldsymbol{e}_{x} + C^{y} \boldsymbol{e}_{y} + C^{z} \boldsymbol{e}_{z} + C^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + C^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + C^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} + C^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \partial_{x} f  \boldsymbol{e}_{x} + \partial_{y} f  \boldsymbol{e}_{y} + \partial_{z} f  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot A = \partial_{x} A^{x}  + \partial_{y} A^{y}  + \partial_{z} A^{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  A = \left ( \partial_{x} A^{x}  + \partial_{y} A^{y}  + \partial_{z} A^{z} \right )  + \left ( - \partial_{y} A^{x}  + \partial_{x} A^{y} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( - \partial_{z} A^{x}  + \partial_{x} A^{z} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( - \partial_{z} A^{y}  + \partial_{y} A^{z} \right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} -I (\boldsymbol{\nabla} \W A) = \left ( - \partial_{z} A^{y}  + \partial_{y} A^{z} \right ) \boldsymbol{e}_{x} + \left ( \partial_{z} A^{x}  - \partial_{x} A^{z} \right ) \boldsymbol{e}_{y} + \left ( - \partial_{y} A^{x}  + \partial_{x} A^{y} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  B = \left ( - \partial_{y} B^{xy}  - \partial_{z} B^{xz} \right ) \boldsymbol{e}_{x} + \left ( \partial_{x} B^{xy}  - \partial_{z} B^{yz} \right ) \boldsymbol{e}_{y} + \left ( \partial_{x} B^{xz}  + \partial_{y} B^{yz} \right ) \boldsymbol{e}_{z} + \left ( \partial_{z} B^{xy}  - \partial_{y} B^{xz}  + \partial_{x} B^{yz} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \W B = \left ( \partial_{z} B^{xy}  - \partial_{y} B^{xz}  + \partial_{x} B^{yz} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot B = \left ( - \partial_{y} B^{xy}  - \partial_{z} B^{xz} \right ) \boldsymbol{e}_{x} + \left ( \partial_{x} B^{xy}  - \partial_{z} B^{yz} \right ) \boldsymbol{e}_{y} + \left ( \partial_{x} B^{xz}  + \partial_{y} B^{yz} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def derivatives_in_spherical_coordinates():
        Print_Function()
        X = (r,th,phi) = symbols('r theta phi')
        s3d = Ga('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=X,norm=True)
        (er,eth,ephi) = s3d.mv()
        grad = s3d.grad
        f = s3d.mv('f','scalar',f=True)
        A = s3d.mv('A','vector',f=True)
        B = s3d.mv('B','bivector',f=True)
        print('f =',f)
        print('A =',A)
        print('B =',B)
        print('grad*f =',grad*f)
        print('grad|A =',grad|A)
        print('-I*(grad^A) =',(-s3d.E()*(grad^A)).simplify())
        print('grad^B =',grad^B)
    \end{lstlisting}
    Code Output:
    \begin{equation*} f = f \end{equation*}
    \begin{equation*} A = A^{r}  \boldsymbol{e}_{r} + A^{\theta }  \boldsymbol{e}_{\theta } + A^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} B = B^{r\theta }  \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\theta } + B^{r\phi }  \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\phi } + B^{\theta \phi }  \boldsymbol{e}_{\theta }\wedge \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \partial_{r} f  \boldsymbol{e}_{r} + \frac{\partial_{\theta } f }{r} \boldsymbol{e}_{\theta } + \frac{\partial_{\phi } f }{r \sin{\left (\theta  \right )}} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot A = \frac{r \partial_{r} A^{r}  + 2 A^{r}  + \frac{A^{\theta } }{\tan{\left (\theta  \right )}} + \partial_{\theta } A^{\theta }  + \frac{\partial_{\phi } A^{\phi } }{\sin{\left (\theta  \right )}}}{r} \end{equation*}
    \begin{equation*} -I (\boldsymbol{\nabla} \W A) = \frac{\frac{A^{\phi } }{\tan{\left (\theta  \right )}} + \partial_{\theta } A^{\phi }  - \frac{\partial_{\phi } A^{\theta } }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{r} + \frac{- r \partial_{r} A^{\phi }  - A^{\phi }  + \frac{\partial_{\phi } A^{r} }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{\theta } + \frac{r \partial_{r} A^{\theta }  + A^{\theta }  - \partial_{\theta } A^{r} }{r} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \W B = \frac{r \partial_{r} B^{\theta \phi }  - \frac{B^{r\phi } }{\tan{\left (\theta  \right )}} + 2 B^{\theta \phi }  - \partial_{\theta } B^{r\phi }  + \frac{\partial_{\phi } B^{r\theta } }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\theta }\wedge \boldsymbol{e}_{\phi } \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def noneuclidian_distance_calculation():
        Print_Function()
        from sympy import solve,sqrt
        Fmt(1)
        g = '0 # #,# 0 #,# # 1'
        nel = Ga('X Y e',g=g)
        (X,Y,e) = nel.mv()
        print('g_{ij} =',nel.g)
        print('%(X\\W Y)^{2} =',(X^Y)*(X^Y))
        L = X^Y^e
        B = L*e # D&L 10.152
        Bsq = (B*B).scalar()
        print('#%L = X\\W Y\\W e \\text{ is a non-euclidian line}')
        print('B = L*e =',B)
        BeBr =B*e*B.rev()
        print('%BeB^{\\dagger} =',BeBr)
        print('%B^{2} =',B*B)
        print('%L^{2} =',L*L) # D&L 10.153
        (s,c,Binv,M,S,C,alpha) = symbols('s c (1/B) M S C alpha')
        XdotY = nel.g[0,1]
        Xdote = nel.g[0,2]
        Ydote = nel.g[1,2]
        Bhat = Binv*B # D&L 10.154
        R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
        print('#%s = \\f{\\sinh}{\\alpha/2} \\text{ and } c = \\f{\\cosh}{\\alpha/2}')
        print('%e^{\\alpha B/{2\\abs{B}}} =',R)
        Z = R*X*R.rev() # D&L 10.155
        Z.obj = expand(Z.obj)
        Z.obj = Z.obj.collect([Binv,s,c,XdotY])
        Z.Fmt(3,'%RXR^{\\dagger}')
        W = Z|Y # Extract scalar part of multivector
        # From this point forward all calculations are with sympy scalars
        #print '#Objective is to determine value of C = cosh(alpha) such that W = 0'
        W = W.scalar()
        print('%W = Z\\cdot Y =',W)
        W = expand(W)
        W = simplify(W)
        W = W.collect([s*Binv])
        M = 1/Bsq
        W = W.subs(Binv**2,M)
        W = simplify(W)
        Bmag = sqrt(XdotY**2-2*XdotY*Xdote*Ydote)
        W = W.collect([Binv*c*s,XdotY])
        #Double angle substitutions
        W = W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2))
        W = W.subs(2*c*s,S)
        W = W.subs(c**2,(C+1)/2)
        W = W.subs(s**2,(C-1)/2)
        W = simplify(W)
        W = W.subs(Binv,1/Bmag)
        W = expand(W)
        print('#%S = \\f{\\sinh}{\\alpha} \\text{ and } C = \\f{\\cosh}{\\alpha}')
        print('W =',W)
        Wd = collect(W,[C,S],exact=True,evaluate=False)
        Wd_1 = Wd[one]
        Wd_C = Wd[C]
        Wd_S = Wd[S]
        print('%\\text{Scalar Coefficient} =',Wd_1)
        print('%\\text{Cosh Coefficient} =',Wd_C)
        print('%\\text{Sinh Coefficient} =',Wd_S)
        print('%\\abs{B} =',Bmag)
        Wd_1 = Wd_1.subs(Bmag,1/Binv)
        Wd_C = Wd_C.subs(Bmag,1/Binv)
        Wd_S = Wd_S.subs(Bmag,1/Binv)
        lhs = Wd_1+Wd_C*C
        rhs = -Wd_S*S
        lhs = lhs**2
        rhs = rhs**2
        W = expand(lhs-rhs)
        W = expand(W.subs(1/Binv**2,Bmag**2))
        W = expand(W.subs(S**2,C**2-1))
        W = W.collect([C,C**2],evaluate=False)
        a = simplify(W[C**2])
        b = simplify(W[C])
        c = simplify(W[one])
        print('#%\\text{Require } aC^{2}+bC+c = 0')
        print('a =',a)
        print('b =',b)
        print('c =',c)
        x = Symbol('x')
        C =  solve(a*x**2+b*x+c,x)[0]
        print('%b^{2}-4ac =',simplify(b**2-4*a*c))
        print('%\\f{\\cosh}{\\alpha} = C = -b/(2a) =',expand(simplify(expand(C))))
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} g_{ij} = \left[\begin{array}{ccc}0 & \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  & \left (\boldsymbol{X}\cdot \boldsymbol{e}\right ) \\\left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  & 0 & \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) \\\left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  & \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right )  & 1\end{array}\right] \end{equation*}
    \begin{equation*} (X\W Y)^{2} = \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right ) ^{2} \end{equation*}
    \begin{equation*} L = X\W Y\W e \text{ is a non-euclidian line} \end{equation*}
    \begin{equation*} B = L e =  \boldsymbol{X}\wedge \boldsymbol{Y} - \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right )  \boldsymbol{X}\wedge \boldsymbol{e} + \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \boldsymbol{Y}\wedge \boldsymbol{e} \end{equation*}
    \begin{equation*} BeB^{\dagger} = \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  \left(- \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  + 2 \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) \right) \boldsymbol{e} \end{equation*}
    \begin{equation*} B^{2} = \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  \left(\left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  - 2 \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) \right) \end{equation*}
    \begin{equation*} L^{2} = \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  \left(\left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  - 2 \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) \right) \end{equation*}
    \begin{equation*} s = \f{\sinh}{\alpha/2} \text{ and } c = \f{\cosh}{\alpha/2} \end{equation*}
    \begin{equation*} e^{\alpha B/{2\abs{B}}} = c  + (1/B) s \boldsymbol{X}\wedge \boldsymbol{Y} - \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right )  (1/B) s \boldsymbol{X}\wedge \boldsymbol{e} + \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  (1/B) s \boldsymbol{Y}\wedge \boldsymbol{e} \end{equation*}
    \begin{equation*} W = Z\cdot Y = \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right ) ^{3} (1/B)^{2} s^{2} - 4 \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right ) ^{2} \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right )  (1/B)^{2} s^{2} + 2 \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right ) ^{2} (1/B) c s + 4 \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  \left (\boldsymbol{X}\cdot \boldsymbol{e}\right ) ^{2} \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) ^{2} (1/B)^{2} s^{2} - 4 \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right )  (1/B) c s + \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  c^{2} \end{equation*}
    \begin{equation*} S = \f{\sinh}{\alpha} \text{ and } C = \f{\cosh}{\alpha} \end{equation*}
    \begin{equation*} W = \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  C - \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right )  C + \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right )  + S \sqrt{\left (\boldsymbol{X}\cdot \boldsymbol{Y}\right ) ^{2} - 2 \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) } \end{equation*}
    \begin{equation*} \text{Scalar Coefficient} = \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) \end{equation*}
    \begin{equation*} \text{Cosh Coefficient} = \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  - \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) \end{equation*}
    \begin{equation*} \text{Sinh Coefficient} = \sqrt{\left (\boldsymbol{X}\cdot \boldsymbol{Y}\right ) ^{2} - 2 \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) } \end{equation*}
    \begin{equation*} \abs{B} = \sqrt{\left (\boldsymbol{X}\cdot \boldsymbol{Y}\right ) ^{2} - 2 \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) } \end{equation*}
    \begin{equation*} \text{Require } aC^{2}+bC+c = 0 \end{equation*}
    \begin{equation*} a = \left (\boldsymbol{X}\cdot \boldsymbol{e}\right ) ^{2} \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) ^{2} \end{equation*}
    \begin{equation*} b = 2 \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right )  \left(\left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  - \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) \right) \end{equation*}
    \begin{equation*} c = \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right ) ^{2} - 2 \left (\boldsymbol{X}\cdot \boldsymbol{Y}\right )  \left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right )  + \left (\boldsymbol{X}\cdot \boldsymbol{e}\right ) ^{2} \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) ^{2} \end{equation*}
    \begin{equation*} b^{2}-4ac = 0 \end{equation*}
    \begin{equation*} \f{\cosh}{\alpha} = C = -b/(2a) = - \frac{\left (\boldsymbol{X}\cdot \boldsymbol{Y}\right ) }{\left (\boldsymbol{X}\cdot \boldsymbol{e}\right )  \left (\boldsymbol{Y}\cdot \boldsymbol{e}\right ) } + 1 \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def conformal_representations_of_circles_lines_spheres_and_planes():
        Print_Function()
        global n,nbar
        Fmt(1)
        g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'
        c3d = Ga('e_1 e_2 e_3 n \\bar{n}',g=g)
        (e1,e2,e3,n,nbar) = c3d.mv()
        print('g_{ij} =',c3d.g)
        e = n+nbar
        #conformal representation of points
        A = make_vector(e1, ga=c3d)    # point a = (1,0,0)  A = F(a)
        B = make_vector(e2, ga=c3d)    # point b = (0,1,0)  B = F(b)
        C = make_vector(-e1, ga=c3d)   # point c = (-1,0,0) C = F(c)
        D = make_vector(e3, ga=c3d)    # point d = (0,0,1)  D = F(d)
        X = make_vector('x',3, ga=c3d)
        print('F(a) =',A)
        print('F(b) =',B)
        print('F(c) =',C)
        print('F(d) =',D)
        print('F(x) =',X)
        print('#a = e1, b = e2, c = -e1, and d = e3')
        print('#A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.')
        print('#Circle through a, b, and c')
        print('Circle: A^B^C^X = 0 =',(A^B^C^X))
        print('#Line through a and b')
        print('Line  : A^B^n^X = 0 =',(A^B^n^X))
        print('#Sphere through a, b, c, and d')
        print('Sphere: A^B^C^D^X = 0 =',(((A^B)^C)^D)^X)
        print('#Plane through a, b, and d')
        print('Plane : A^B^n^D^X = 0 =',(A^B^n^D^X))
        L = (A^B^e)^X
        L.Fmt(3,'Hyperbolic\\;\\; Circle: (A^B^e)^X = 0')
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} g_{ij} = \left[\begin{array}{ccccc}1 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 2\\0 & 0 & 0 & 2 & 0\end{array}\right] \end{equation*}
    \begin{equation*} F(a) =  \boldsymbol{e}_{1} + \frac{1}{2} \boldsymbol{n} - \frac{1}{2} \boldsymbol{\bar{n}} \end{equation*}
    \begin{equation*} F(b) =  \boldsymbol{e}_{2} + \frac{1}{2} \boldsymbol{n} - \frac{1}{2} \boldsymbol{\bar{n}} \end{equation*}
    \begin{equation*} F(c) = - \boldsymbol{e}_{1} + \frac{1}{2} \boldsymbol{n} - \frac{1}{2} \boldsymbol{\bar{n}} \end{equation*}
    \begin{equation*} F(d) =  \boldsymbol{e}_{3} + \frac{1}{2} \boldsymbol{n} - \frac{1}{2} \boldsymbol{\bar{n}} \end{equation*}
    \begin{equation*} F(x) = x_{1} \boldsymbol{e}_{1} + x_{2} \boldsymbol{e}_{2} + x_{3} \boldsymbol{e}_{3} + \left ( \frac{{\left ( x_{1} \right )}^{2}}{2} + \frac{{\left ( x_{2} \right )}^{2}}{2} + \frac{{\left ( x_{3} \right )}^{2}}{2}\right ) \boldsymbol{n} - \frac{1}{2} \boldsymbol{\bar{n}} \end{equation*}
    a = e1, b = e2, c = -e1, and d = e3
    A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.
    Circle through a, b, and c
    \begin{equation*} Circle: A\W B\W C\W X = 0 = - x_{3} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{n} + x_{3} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{\bar{n}} + \left ( \frac{{\left ( x_{1} \right )}^{2}}{2} + \frac{{\left ( x_{2} \right )}^{2}}{2} + \frac{{\left ( x_{3} \right )}^{2}}{2} - \frac{1}{2}\right ) \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{n}\wedge \boldsymbol{\bar{n}} \end{equation*}
    Line through a and b
    \begin{equation*} Line  : A\W B\W n\W X = 0 = - x_{3} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{n} + \left ( \frac{x_{1}}{2} + \frac{x_{2}}{2} - \frac{1}{2}\right ) \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{n}\wedge \boldsymbol{\bar{n}} + \frac{x_{3}}{2} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{n}\wedge \boldsymbol{\bar{n}} - \frac{x_{3}}{2} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{n}\wedge \boldsymbol{\bar{n}} \end{equation*}
    Sphere through a, b, c, and d
    \begin{equation*} Sphere: A\W B\W C\W D\W X = 0 = \left ( - \frac{{\left ( x_{1} \right )}^{2}}{2} - \frac{{\left ( x_{2} \right )}^{2}}{2} - \frac{{\left ( x_{3} \right )}^{2}}{2} + \frac{1}{2}\right ) \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{n}\wedge \boldsymbol{\bar{n}} \end{equation*}
    Plane through a, b, and d
    \begin{equation*} Plane : A\W B\W n\W D\W X = 0 = \left ( - \frac{x_{1}}{2} - \frac{x_{2}}{2} - \frac{x_{3}}{2} + \frac{1}{2}\right ) \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{n}\wedge \boldsymbol{\bar{n}} \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def properties_of_geometric_objects():
        Print_Function()
        global n, nbar
        Fmt(1)
        g = '# # # 0 0,'+ \
            '# # # 0 0,'+ \
            '# # # 0 0,'+ \
            '0 0 0 0 2,'+ \
            '0 0 0 2 0'
        c3d = Ga('p1 p2 p3 n \\bar{n}',g=g)
        (p1,p2,p3,n,nbar) = c3d.mv()
        print('g_{ij} =',c3d.g)
        P1 = F(p1)
        P2 = F(p2)
        P3 = F(p3)
        print('\\text{Extracting direction of line from }L = P1\\W P2\\W n')
        L = P1^P2^n
        delta = (L|n)|nbar
        print('(L|n)|\\bar{n} =',delta)
        print('\\text{Extracting plane of circle from }C = P1\\W P2\\W P3')
        C = P1^P2^P3
        delta = ((C^n)|n)|nbar
        print('((C^n)|n)|\\bar{n}=',delta)
        print('(p2-p1)^(p3-p1)=',(p2-p1)^(p3-p1))
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} g_{ij} = \left[\begin{array}{ccccc}\left (\boldsymbol{p}_{1}\cdot \boldsymbol{p}_{1}\right )  & \left (\boldsymbol{p}_{1}\cdot \boldsymbol{p}_{2}\right )  & \left (\boldsymbol{p}_{1}\cdot \boldsymbol{p}_{3}\right )  & 0 & 0\\\left (\boldsymbol{p}_{1}\cdot \boldsymbol{p}_{2}\right )  & \left (\boldsymbol{p}_{2}\cdot \boldsymbol{p}_{2}\right )  & \left (\boldsymbol{p}_{2}\cdot \boldsymbol{p}_{3}\right )  & 0 & 0\\\left (\boldsymbol{p}_{1}\cdot \boldsymbol{p}_{3}\right )  & \left (\boldsymbol{p}_{2}\cdot \boldsymbol{p}_{3}\right )  & \left (\boldsymbol{p}_{3}\cdot \boldsymbol{p}_{3}\right )  & 0 & 0\\0 & 0 & 0 & 0 & 2\\0 & 0 & 0 & 2 & 0\end{array}\right] \end{equation*}
    \begin{equation*} \text{Extracting direction of line from }L = P1\W P2\W n \end{equation*}
    \begin{equation*} (L\cdot n)\cdot \bar{n} = 2 \boldsymbol{p}_{1} -2 \boldsymbol{p}_{2} \end{equation*}
    \begin{equation*} \text{Extracting plane of circle from }C = P1\W P2\W P3 \end{equation*}
    \begin{equation*} ((C\W n)\cdot n)\cdot \bar{n}= 2 \boldsymbol{p}_{1}\wedge \boldsymbol{p}_{2} -2 \boldsymbol{p}_{1}\wedge \boldsymbol{p}_{3} + 2 \boldsymbol{p}_{2}\wedge \boldsymbol{p}_{3} \end{equation*}
    \begin{equation*} (p2-p1)\W (p3-p1)=  \boldsymbol{p}_{1}\wedge \boldsymbol{p}_{2} - \boldsymbol{p}_{1}\wedge \boldsymbol{p}_{3} + \boldsymbol{p}_{2}\wedge \boldsymbol{p}_{3} \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def extracting_vectors_from_conformal_2_blade():
        Print_Function()
        Fmt(1)
        print(r'B = P1\W P2')
        g = '0 -1 #,'+ \
            '-1 0 #,'+ \
            '# # #'
        c2b = Ga('P1 P2 a',g=g)
        (P1,P2,a) = c2b.mv()
        print('g_{ij} =',c2b.g)
        B = P1^P2
        Bsq = B*B
        print('%B^{2} =',Bsq)
        ap = a-(a^B)*B
        print("a' = a-(a^B)*B =",ap)
        Ap = ap+ap*B
        Am = ap-ap*B
        print("A+ = a'+a'*B =",Ap)
        print("A- = a'-a'*B =",Am)
        print('%(A+)^{2} =',Ap*Ap)
        print('%(A-)^{2} =',Am*Am)
        aB = a|B
        print('a|B =',aB)
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} B = P1\W P2 \end{equation*}
    \begin{equation*} g_{ij} = \left[\begin{array}{ccc}0 & -1 & \left (\boldsymbol{P}_{1}\cdot \boldsymbol{a}\right ) \\-1 & 0 & \left (\boldsymbol{P}_{2}\cdot \boldsymbol{a}\right ) \\\left (\boldsymbol{P}_{1}\cdot \boldsymbol{a}\right )  & \left (\boldsymbol{P}_{2}\cdot \boldsymbol{a}\right )  & \left (\boldsymbol{a}\cdot \boldsymbol{a}\right ) \end{array}\right] \end{equation*}
    \begin{equation*} B^{2} = 1 \end{equation*}
    \begin{equation*} a' = a-(a\W B) B = - \left (\boldsymbol{P}_{2}\cdot \boldsymbol{a}\right )  \boldsymbol{P}_{1} - \left (\boldsymbol{P}_{1}\cdot \boldsymbol{a}\right )  \boldsymbol{P}_{2} \end{equation*}
    \begin{equation*} A+ = a'+a' B = - 2 \left (\boldsymbol{P}_{2}\cdot \boldsymbol{a}\right )  \boldsymbol{P}_{1} \end{equation*}
    \begin{equation*} A- = a'-a' B = - 2 \left (\boldsymbol{P}_{1}\cdot \boldsymbol{a}\right )  \boldsymbol{P}_{2} \end{equation*}
    \begin{equation*} (A+)^{2} =  0 \end{equation*}
    \begin{equation*} (A-)^{2} =  0 \end{equation*}
    \begin{equation*} a\cdot B = - \left (\boldsymbol{P}_{2}\cdot \boldsymbol{a}\right )  \boldsymbol{P}_{1} + \left (\boldsymbol{P}_{1}\cdot \boldsymbol{a}\right )  \boldsymbol{P}_{2} \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def reciprocal_frame_test():
        Print_Function()
        Fmt(1)
        g = '1 # #,'+ \
            '# 1 #,'+ \
            '# # 1'
        ng3d = Ga('e1 e2 e3',g=g)
        (e1,e2,e3) = ng3d.mv()
        print('g_{ij} =',ng3d.g)
        E = e1^e2^e3
        Esq = (E*E).scalar()
        print('E =',E)
        print('%E^{2} =',Esq)
        Esq_inv = 1/Esq
        E1 = (e2^e3)*E
        E2 = (-1)*(e1^e3)*E
        E3 = (e1^e2)*E
        print('E1 = (e2^e3)*E =',E1)
        print('E2 =-(e1^e3)*E =',E2)
        print('E3 = (e1^e2)*E =',E3)
        w = (E1|e2)
        w = w.expand()
        print('E1|e2 =',w)
        w = (E1|e3)
        w = w.expand()
        print('E1|e3 =',w)
        w = (E2|e1)
        w = w.expand()
        print('E2|e1 =',w)
        w = (E2|e3)
        w = w.expand()
        print('E2|e3 =',w)
        w = (E3|e1)
        w = w.expand()
        print('E3|e1 =',w)
        w = (E3|e2)
        w = w.expand()
        print('E3|e2 =',w)
        w = (E1|e1)
        w = (w.expand()).scalar()
        Esq = expand(Esq)
        print('%(E1\\cdot e1)/E^{2} =',simplify(w/Esq))
        w = (E2|e2)
        w = (w.expand()).scalar()
        print('%(E2\\cdot e2)/E^{2} =',simplify(w/Esq))
        w = (E3|e3)
        w = (w.expand()).scalar()
        print('%(E3\\cdot e3)/E^{2} =',simplify(w/Esq))
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} g_{ij} = \left[\begin{array}{ccc}1 & \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  & \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right ) \\\left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  & 1 & \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) \\\left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  & \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right )  & 1\end{array}\right] \end{equation*}
    \begin{equation*} E =  \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} E^{2} = \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right ) ^{2} - 2 \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right ) ^{2} + \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) ^{2} - 1 \end{equation*}
    \begin{equation*} E1 = (e2\W e3) E = \left ( \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) ^{2} - 1\right ) \boldsymbol{e}_{1} + \left ( \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{2} + \left ( - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} E2 =-(e1\W e3) E = \left ( \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{1} + \left ( \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right ) ^{2} - 1\right ) \boldsymbol{e}_{2} + \left ( - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} E3 = (e1\W e2) E = \left ( - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{1} + \left ( - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{2} + \left ( \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right ) ^{2} - 1\right ) \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} E1\cdot e2 =  0 \end{equation*}
    \begin{equation*} E1\cdot e3 =  0 \end{equation*}
    \begin{equation*} E2\cdot e1 =  0 \end{equation*}
    \begin{equation*} E2\cdot e3 =  0 \end{equation*}
    \begin{equation*} E3\cdot e1 =  0 \end{equation*}
    \begin{equation*} E3\cdot e2 =  0 \end{equation*}
    \begin{equation*} (E1\cdot e1)/E^{2} = 1 \end{equation*}
    \begin{equation*} (E2\cdot e2)/E^{2} = 1 \end{equation*}
    \begin{equation*} (E3\cdot e3)/E^{2} = 1 \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def signature_test():
        Print_Function()
        e3d = Ga('e1 e2 e3',g=[1,1,1])
        print('g =', e3d.g)
        print(r'%Signature = (3,0)\: I =', e3d.I(), r'\: I^{2} =', e3d.I()*e3d.I())
        e3d = Ga('e1 e2 e3',g=[2,2,2])
        print('g =', e3d.g)
        print(r'%Signature = (3,0)\: I =', e3d.I(),'|; I^{2} =', e3d.I()*e3d.I())
        sp4d = Ga('e1 e2 e3 e4',g=[1,-1,-1,-1])
        print('g =', sp4d.g)
        print(r'%Signature = (1,3)\: I =', sp4d.I(), r'\: I^{2} =', sp4d.I()*sp4d.I())
        sp4d = Ga('e1 e2 e3 e4',g=[2,-2,-2,-2])
        print('g =', sp4d.g)
        print(r'%Signature = (1,3)\: I =', sp4d.I(), r'\: I^{2} =', sp4d.I()*sp4d.I())
        e4d = Ga('e1 e2 e3 e4',g=[1,1,1,1])
        print('g =', e4d.g)
        print(r'%Signature = (4,0)\: I =', e4d.I(), r'\: I^{2} =', e4d.I()*e4d.I())
        cf3d = Ga('e1 e2 e3 e4 e5',g=[1,1,1,1,-1])
        print('g =', cf3d.g)
        print(r'%Signature = (4,1)\: I =', cf3d.I(), r'\: I^{2} =', cf3d.I()*cf3d.I())
        cf3d = Ga('e1 e2 e3 e4 e5',g=[2,2,2,2,-2])
        print('g =', cf3d.g)
        print(r'%Signature = (4,1)\: I =', cf3d.I(), r'\: I^{2} =', cf3d.I()*cf3d.I())
        return
    \end{lstlisting}
    Code Output:
    \begin{equation*} g = \left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right] \end{equation*}
    \begin{equation*} Signature = (3,0)\: I =  \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} \: I^{2} = -1 \end{equation*}
    \begin{equation*} g = \left[\begin{array}{ccc}2 & 0 & 0\\0 & 2 & 0\\0 & 0 & 2\end{array}\right] \end{equation*}
    \begin{equation*} Signature = (3,0)\: I = \frac{\sqrt{2}}{4} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} |; I^{2} = -1 \end{equation*}
    \begin{equation*} g = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & -1\end{array}\right] \end{equation*}
    \begin{equation*} Signature = (1,3)\: I =  \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{4} \: I^{2} = -1 \end{equation*}
    \begin{equation*} g = \left[\begin{array}{cccc}2 & 0 & 0 & 0\\0 & -2 & 0 & 0\\0 & 0 & -2 & 0\\0 & 0 & 0 & -2\end{array}\right] \end{equation*}
    \begin{equation*} Signature = (1,3)\: I = \frac{1}{4} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{4} \: I^{2} = -1 \end{equation*}
    \begin{equation*} g = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{array}\right] \end{equation*}
    \begin{equation*} Signature = (4,0)\: I =  \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{4} \: I^{2} = 1 \end{equation*}
    \begin{equation*} g = \left[\begin{array}{ccccc}1 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & -1\end{array}\right] \end{equation*}
    \begin{equation*} Signature = (4,1)\: I =  \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{4}\wedge \boldsymbol{e}_{5} \: I^{2} = -1 \end{equation*}
    \begin{equation*} g = \left[\begin{array}{ccccc}2 & 0 & 0 & 0 & 0\\0 & 2 & 0 & 0 & 0\\0 & 0 & 2 & 0 & 0\\0 & 0 & 0 & 2 & 0\\0 & 0 & 0 & 0 & -2\end{array}\right] \end{equation*}
    \begin{equation*} Signature = (4,1)\: I = \frac{\sqrt{2}}{8} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{4}\wedge \boldsymbol{e}_{5} \: I^{2} = -1 \end{equation*}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def Fmt_test():
        Print_Function()
        e3d = Ga('e1 e2 e3',g=[1,1,1])
        v = e3d.mv('v','vector')
        B = e3d.mv('B','bivector')
        M = e3d.mv('M','mv')
        Fmt(2)
        print('#Global $Fmt = 2$')
        print('v =',v)
        print('B =',B)
        print('M =',M)
        print('#Using $.Fmt()$ Function')
        print('v.Fmt(3) =',v.Fmt(3))
        print('B.Fmt(3) =',B.Fmt(3))
        print('M.Fmt(2) =',M.Fmt(2))
        print('M.Fmt(1) =',M.Fmt(1))
        print('#Global $Fmt = 1$')
        Fmt(1)
        print('v =',v)
        print('B =',B)
        print('M =',M)
        return
    \end{lstlisting}
    Code Output:
    Global $Fmt = 2$
    \begin{equation*} v = v^{1} \boldsymbol{e}_{1} + v^{2} \boldsymbol{e}_{2} + v^{3} \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} B = B^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} + B^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} + B^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} M =  \begin{aligned}[t]  & M  \\  &  + M^{1} \boldsymbol{e}_{1} + M^{2} \boldsymbol{e}_{2} + M^{3} \boldsymbol{e}_{3} \\  &  + M^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} + M^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} + M^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} \\  &  + M^{123} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}  \end{aligned} \end{equation*}
    Using $.Fmt()$ Function
    \begin{equation*} v.Fmt(3) =  \begin{aligned}[t]  & v^{1} \boldsymbol{e}_{1} \\  &  + v^{2} \boldsymbol{e}_{2} \\  &  + v^{3} \boldsymbol{e}_{3}  \end{aligned} \end{equation*}
    \begin{equation*} B.Fmt(3) =  \begin{aligned}[t]  & B^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} \\  &  + B^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} \\  &  + B^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}  \end{aligned} \end{equation*}
    \begin{equation*} M.Fmt(2) =  \begin{aligned}[t]  & M  \\  &  + M^{1} \boldsymbol{e}_{1} + M^{2} \boldsymbol{e}_{2} + M^{3} \boldsymbol{e}_{3} \\  &  + M^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} + M^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} + M^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} \\  &  + M^{123} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}  \end{aligned} \end{equation*}
    \begin{equation*} M.Fmt(1) = M  + M^{1} \boldsymbol{e}_{1} + M^{2} \boldsymbol{e}_{2} + M^{3} \boldsymbol{e}_{3} + M^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} + M^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} + M^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} + M^{123} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} \end{equation*}
    Global $Fmt = 1$
    \begin{equation*} v = v^{1} \boldsymbol{e}_{1} + v^{2} \boldsymbol{e}_{2} + v^{3} \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} B = B^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} + B^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} + B^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} M = M  + M^{1} \boldsymbol{e}_{1} + M^{2} \boldsymbol{e}_{2} + M^{3} \boldsymbol{e}_{3} + M^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} + M^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} + M^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} + M^{123} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} \end{equation*}
    \end{document}

```python
check('lin_tran_check')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    3d orthogonal ($A$ is vector function)
    \begin{equation*} A = A^{x}  \boldsymbol{e}_{x} + A^{y}  \boldsymbol{e}_{y} + A^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} A^{2} = {A^{x} }^{2} + {A^{y} }^{2} + {A^{z} }^{2} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot A = \partial_{x} A^{x}  + \partial_{y} A^{y}  + \partial_{z} A^{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  A = \left ( \partial_{x} A^{x}  + \partial_{y} A^{y}  + \partial_{z} A^{z} \right )  + \left ( - \partial_{y} A^{x}  + \partial_{x} A^{y} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( - \partial_{z} A^{x}  + \partial_{x} A^{z} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( - \partial_{z} A^{y}  + \partial_{y} A^{z} \right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v\cdot (\boldsymbol{\nabla}  A) = \left ( v^{y} \partial_{y} A^{x}  - v^{y} \partial_{x} A^{y}  + v^{z} \partial_{z} A^{x}  - v^{z} \partial_{x} A^{z} \right ) \boldsymbol{e}_{x} + \left ( - v^{x} \partial_{y} A^{x}  + v^{x} \partial_{x} A^{y}  + v^{z} \partial_{z} A^{y}  - v^{z} \partial_{y} A^{z} \right ) \boldsymbol{e}_{y} + \left ( - v^{x} \partial_{z} A^{x}  + v^{x} \partial_{x} A^{z}  - v^{y} \partial_{z} A^{y}  + v^{y} \partial_{y} A^{z} \right ) \boldsymbol{e}_{z} \end{equation*}
    2d general ($A$ is vector function)
    \begin{equation*} A = A^{u}  \boldsymbol{e}_{u} + A^{v}  \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} A^{2} = \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  {A^{u} }^{2} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A^{u}  A^{v}  + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  {A^{v} }^{2} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot A = \partial_{u} A^{u}  + \partial_{v} A^{v} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  A = \left ( \partial_{u} A^{u}  + \partial_{v} A^{v} \right )  + \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \partial_{v} A^{u}  + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \partial_{u} A^{u}  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \partial_{v} A^{v}  + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  \partial_{u} A^{v} }{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{v} \end{equation*}
    3d orthogonal ($A,\;B$ are linear transformations)
    \begin{equation*} A = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{x}\right ) =& A_{xx} \boldsymbol{e}_{x} + A_{yx} \boldsymbol{e}_{y} + A_{zx} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{y}\right ) =& A_{xy} \boldsymbol{e}_{x} + A_{yy} \boldsymbol{e}_{y} + A_{zy} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{z}\right ) =& A_{xz} \boldsymbol{e}_{x} + A_{yz} \boldsymbol{e}_{y} + A_{zz} \boldsymbol{e}_{z}  \end{array} \right \} \end{equation*}
    \begin{equation*} \f{mat}{A} = \left[\begin{array}{ccc}A_{xx} & A_{xy} & A_{xz}\\A_{yx} & A_{yy} & A_{yz}\\A_{zx} & A_{zy} & A_{zz}\end{array}\right] \end{equation*}
    \begin{equation*} \f{\det}{A} = A_{xz} \left(A_{yx} A_{zy} - A_{yy} A_{zx}\right) - A_{yz} \left(A_{xx} A_{zy} - A_{xy} A_{zx}\right) + A_{zz} \left(A_{xx} A_{yy} - A_{xy} A_{yx}\right) \end{equation*}
    \begin{equation*} \overline{A} = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{x}\right ) =& A_{xx} \boldsymbol{e}_{x} + A_{xy} \boldsymbol{e}_{y} + A_{xz} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{y}\right ) =& A_{yx} \boldsymbol{e}_{x} + A_{yy} \boldsymbol{e}_{y} + A_{yz} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{z}\right ) =& A_{zx} \boldsymbol{e}_{x} + A_{zy} \boldsymbol{e}_{y} + A_{zz} \boldsymbol{e}_{z}  \end{array} \right \} \end{equation*}
    \begin{equation*} \f{\Tr}{A} = A_{xx} + A_{yy} + A_{zz} \end{equation*}
    \begin{equation*} \f{A}{e_x\W e_y} = \left ( A_{xx} A_{yy} - A_{xy} A_{yx}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( A_{xx} A_{zy} - A_{xy} A_{zx}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( A_{yx} A_{zy} - A_{yy} A_{zx}\right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \f{A}{e_x}\W \f{A}{e_y} = \left ( A_{xx} A_{yy} - A_{xy} A_{yx}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( A_{xx} A_{zy} - A_{xy} A_{zx}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( A_{yx} A_{zy} - A_{yy} A_{zx}\right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} g = \left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right] \end{equation*}
    \begin{equation*} g^{-1} = \left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right] \end{equation*}
    \begin{equation*} A + B = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{x}\right ) =& \left ( A_{xx} + B_{xx}\right ) \boldsymbol{e}_{x} + \left ( A_{yx} + B_{yx}\right ) \boldsymbol{e}_{y} + \left ( A_{zx} + B_{zx}\right ) \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{y}\right ) =& \left ( A_{xy} + B_{xy}\right ) \boldsymbol{e}_{x} + \left ( A_{yy} + B_{yy}\right ) \boldsymbol{e}_{y} + \left ( A_{zy} + B_{zy}\right ) \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{z}\right ) =& \left ( A_{xz} + B_{xz}\right ) \boldsymbol{e}_{x} + \left ( A_{yz} + B_{yz}\right ) \boldsymbol{e}_{y} + \left ( A_{zz} + B_{zz}\right ) \boldsymbol{e}_{z}  \end{array} \right \} \end{equation*}
    \begin{equation*} AB = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{x}\right ) =& \left ( A_{xx} B_{xx} + A_{xy} B_{yx} + A_{xz} B_{zx}\right ) \boldsymbol{e}_{x} + \left ( A_{yx} B_{xx} + A_{yy} B_{yx} + A_{yz} B_{zx}\right ) \boldsymbol{e}_{y} + \left ( A_{zx} B_{xx} + A_{zy} B_{yx} + A_{zz} B_{zx}\right ) \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{y}\right ) =& \left ( A_{xx} B_{xy} + A_{xy} B_{yy} + A_{xz} B_{zy}\right ) \boldsymbol{e}_{x} + \left ( A_{yx} B_{xy} + A_{yy} B_{yy} + A_{yz} B_{zy}\right ) \boldsymbol{e}_{y} + \left ( A_{zx} B_{xy} + A_{zy} B_{yy} + A_{zz} B_{zy}\right ) \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{z}\right ) =& \left ( A_{xx} B_{xz} + A_{xy} B_{yz} + A_{xz} B_{zz}\right ) \boldsymbol{e}_{x} + \left ( A_{yx} B_{xz} + A_{yy} B_{yz} + A_{yz} B_{zz}\right ) \boldsymbol{e}_{y} + \left ( A_{zx} B_{xz} + A_{zy} B_{yz} + A_{zz} B_{zz}\right ) \boldsymbol{e}_{z}  \end{array} \right \} \end{equation*}
    \begin{equation*} A - B = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{x}\right ) =& \left ( A_{xx} - B_{xx}\right ) \boldsymbol{e}_{x} + \left ( A_{yx} - B_{yx}\right ) \boldsymbol{e}_{y} + \left ( A_{zx} - B_{zx}\right ) \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{y}\right ) =& \left ( A_{xy} - B_{xy}\right ) \boldsymbol{e}_{x} + \left ( A_{yy} - B_{yy}\right ) \boldsymbol{e}_{y} + \left ( A_{zy} - B_{zy}\right ) \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{z}\right ) =& \left ( A_{xz} - B_{xz}\right ) \boldsymbol{e}_{x} + \left ( A_{yz} - B_{yz}\right ) \boldsymbol{e}_{y} + \left ( A_{zz} - B_{zz}\right ) \boldsymbol{e}_{z}  \end{array} \right \} \end{equation*}
    \begin{equation*} General Symmetric Linear Transformation \end{equation*}
    \begin{equation*} A = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{x}\right ) =& A_{xx} \boldsymbol{e}_{x} + A_{xy} \boldsymbol{e}_{y} + A_{xz} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{y}\right ) =& A_{xy} \boldsymbol{e}_{x} + A_{yy} \boldsymbol{e}_{y} + A_{yz} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{z}\right ) =& A_{xz} \boldsymbol{e}_{x} + A_{yz} \boldsymbol{e}_{y} + A_{zz} \boldsymbol{e}_{z}  \end{array} \right \} \end{equation*}
    \begin{equation*} General Antisymmetric Linear Transformation \end{equation*}
    \begin{equation*} A = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{x}\right ) =& - A_{xy} \boldsymbol{e}_{y} - A_{xz} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{y}\right ) =& A_{xy} \boldsymbol{e}_{x} - A_{yz} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{z}\right ) =& A_{xz} \boldsymbol{e}_{x} + A_{yz} \boldsymbol{e}_{y}  \end{array} \right \} \end{equation*}
    2d general ($A,\;B$ are linear transformations)
    \begin{equation*} g = \left[\begin{array}{cc}\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  & \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) \\\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  & \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) \end{array}\right] \end{equation*}
    \begin{equation*} g^{-1} = \left[\begin{array}{cc}\frac{\left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) }{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} & - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) }{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\\- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) }{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} & \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) }{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\end{array}\right] \end{equation*}
    \begin{equation*} gg^{-1} = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right] \end{equation*}
    \begin{equation*} A = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{u}\right ) =& \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u} + \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{v} \\ L \left ( \boldsymbol{e}_{v}\right ) =& \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{v}  \end{array} \right \} \end{equation*}
    \begin{equation*} \f{mat}{A} = \left[\begin{array}{cc}A_{uu} & A_{uv}\\A_{vu} & A_{vv}\end{array}\right] \end{equation*}
    \begin{equation*} \f{\det}{A} = \frac{- \left(\frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{uv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \left(- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} + \frac{\left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{v} + \left(\frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{vv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \left(- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} + \frac{\left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{v}}{\sqrt{\left (\left(\frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{uv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{u} + \left(\frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{vv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{v}\cdot \left(\frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{uv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{u} + \left(\frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{vv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{v}\right )  \left (\left(- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} + \frac{\left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{u} + \left(- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} + \frac{\left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{v}\cdot \left(- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} + \frac{\left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{u} + \left(- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} + \frac{\left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{v}\right )  - \left (\left(- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} + \frac{\left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{u} + \left(- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} + \frac{\left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{v}\cdot \left(\frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{uv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{u} + \left(\frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{vv}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}}\right) \boldsymbol{e}_{v}\right ) ^{2}}} \end{equation*}
    \begin{equation*} \overline{A} = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{u}\right ) =& \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} \boldsymbol{e}_{u} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{3} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} \boldsymbol{e}_{v} \\ L \left ( \boldsymbol{e}_{v}\right ) =& \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} \boldsymbol{e}_{u} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} \boldsymbol{e}_{v}  \end{array} \right \} \end{equation*}
    \begin{equation*} \f{mat}{\overline{A}} = \left[\begin{array}{cc}- \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left(\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vu}\right) + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left(\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vv} - \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vu}\right)}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} & \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vv} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4} A_{vu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uv} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{4} A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}}\\\frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{4} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{3} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{3} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vv} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} & \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left(\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{3} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu}\right) + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  \left(\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vu}\right)}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} - 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}}\end{array}\right] \end{equation*}
    \begin{equation*} \f{\Tr}{A} = - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vv}}{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{vv}}{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uv}}{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu}}{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} A_{uu}}{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{uv}}{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} - \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{3} A_{vu}}{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu}}{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right ) ^{2} + 2 \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2} \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{4}} \end{equation*}
    \begin{equation*} \f{A}{e_u\W e_v} = \frac{A_{uu} A_{vv} - A_{uv} A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} \f{A}{e_u}\W \f{A}{e_v} = \frac{A_{uu} A_{vv} - A_{uv} A_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} B = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{u}\right ) =& \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{uv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  B_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u} + \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{vv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  B_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{v} \\ L \left ( \boldsymbol{e}_{v}\right ) =& \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  B_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  B_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{v}  \end{array} \right \} \end{equation*}
    \begin{equation*} A + B = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{u}\right ) =& \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{uv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  B_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u} + \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{vv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  B_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{v} \\ L \left ( \boldsymbol{e}_{v}\right ) =& \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  B_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{vv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  B_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vu} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{v}  \end{array} \right \} \end{equation*}
    \begin{equation*} A - B = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{u}\right ) =& \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{uv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{uu} - \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  B_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u} + \frac{- \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vv} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{vv} + \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  A_{vu} - \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  B_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{v} \\ L \left ( \boldsymbol{e}_{v}\right ) =& \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  B_{uv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{uu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{uu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{u} + \frac{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  A_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  B_{vv} - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  A_{vu} + \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right )  B_{vu}}{\left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{u}\right )  \left (\boldsymbol{e}_{v}\cdot \boldsymbol{e}_{v}\right )  - \left (\boldsymbol{e}_{u}\cdot \boldsymbol{e}_{v}\right ) ^{2}} \boldsymbol{e}_{v}  \end{array} \right \} \end{equation*}
    \begin{equation*} a\cdot \f{\overline{A}}{b}-b\cdot \f{\underline{A}}{a} =  0 \end{equation*}
    \begin{equation*} g = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & -1\end{array}\right] \end{equation*}
    \begin{equation*} \underline{T} = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{t}\right ) =& T_{tt} \boldsymbol{e}_{t} + T_{xt} \boldsymbol{e}_{x} + T_{yt} \boldsymbol{e}_{y} + T_{zt} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{x}\right ) =& - T_{tx} \boldsymbol{e}_{t} - T_{xx} \boldsymbol{e}_{x} - T_{yx} \boldsymbol{e}_{y} - T_{zx} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{y}\right ) =& - T_{ty} \boldsymbol{e}_{t} - T_{xy} \boldsymbol{e}_{x} - T_{yy} \boldsymbol{e}_{y} - T_{zy} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{z}\right ) =& - T_{tz} \boldsymbol{e}_{t} - T_{xz} \boldsymbol{e}_{x} - T_{yz} \boldsymbol{e}_{y} - T_{zz} \boldsymbol{e}_{z}  \end{array} \right \} \end{equation*}
    \begin{equation*} \overline{T} = \left \{ \begin{array}{ll} L \left ( \boldsymbol{e}_{t}\right ) =& T_{tt} \boldsymbol{e}_{t} + T_{tx} \boldsymbol{e}_{x} + T_{ty} \boldsymbol{e}_{y} + T_{tz} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{x}\right ) =& - T_{xt} \boldsymbol{e}_{t} - T_{xx} \boldsymbol{e}_{x} - T_{xy} \boldsymbol{e}_{y} - T_{xz} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{y}\right ) =& - T_{yt} \boldsymbol{e}_{t} - T_{yx} \boldsymbol{e}_{x} - T_{yy} \boldsymbol{e}_{y} - T_{yz} \boldsymbol{e}_{z} \\ L \left ( \boldsymbol{e}_{z}\right ) =& - T_{zt} \boldsymbol{e}_{t} - T_{zx} \boldsymbol{e}_{x} - T_{zy} \boldsymbol{e}_{y} - T_{zz} \boldsymbol{e}_{z}  \end{array} \right \} \end{equation*}
    \begin{equation*} \f{\det}{\underline{T}} = T_{tz} \left(T_{xt} T_{yx} T_{zy} - T_{xt} T_{yy} T_{zx} - T_{xx} T_{yt} T_{zy} + T_{xx} T_{yy} T_{zt} + T_{xy} T_{yt} T_{zx} - T_{xy} T_{yx} T_{zt}\right) - T_{xz} \left(T_{tt} T_{yx} T_{zy} - T_{tt} T_{yy} T_{zx} - T_{tx} T_{yt} T_{zy} + T_{tx} T_{yy} T_{zt} + T_{ty} T_{yt} T_{zx} - T_{ty} T_{yx} T_{zt}\right) + T_{yz} \left(T_{tt} T_{xx} T_{zy} - T_{tt} T_{xy} T_{zx} - T_{tx} T_{xt} T_{zy} + T_{tx} T_{xy} T_{zt} + T_{ty} T_{xt} T_{zx} - T_{ty} T_{xx} T_{zt}\right) - T_{zz} \left(T_{tt} T_{xx} T_{yy} - T_{tt} T_{xy} T_{yx} - T_{tx} T_{xt} T_{yy} + T_{tx} T_{xy} T_{yt} + T_{ty} T_{xt} T_{yx} - T_{ty} T_{xx} T_{yt}\right) \end{equation*}
    \begin{equation*} \f{\mbox{tr}}{\underline{T}} = T_{tt} - T_{xx} - T_{yy} - T_{zz} \end{equation*}
    \begin{equation*} a\cdot \f{\overline{T}}{b}-b\cdot \f{\underline{T}}{a} =  0 \end{equation*}
    \begin{equation*} f = f \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \partial_{u} f  \boldsymbol{e}_{u} + \frac{\partial_{v} f }{\sin{\left (u \right )}} \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} F = F^{u}  \boldsymbol{e}_{u} + F^{v}  \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  F = \left ( \frac{F^{u} }{\tan{\left (u \right )}} + \partial_{u} F^{u}  + \frac{\partial_{v} F^{v} }{\sin{\left (u \right )}}\right )  + \left ( \frac{F^{v} }{\tan{\left (u \right )}} + \partial_{u} F^{v}  - \frac{\partial_{v} F^{u} }{\sin{\left (u \right )}}\right ) \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} f = f \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \partial_{\theta } f  \boldsymbol{e}_{\theta } + \frac{\partial_{\phi } f }{\sin{\left (\theta  \right )}} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} F = F^{\theta }  \boldsymbol{e}_{\theta } + F^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  F = \left ( \frac{F^{\theta } }{\tan{\left (\theta  \right )}} + \partial_{\theta } F^{\theta }  + \frac{\partial_{\phi } F^{\phi } }{\sin{\left (\theta  \right )}}\right )  + \left ( \frac{F^{\phi } }{\tan{\left (\theta  \right )}} + \partial_{\theta } F^{\phi }  - \frac{\partial_{\phi } F^{\theta } }{\sin{\left (\theta  \right )}}\right ) \boldsymbol{e}_{\theta }\wedge \boldsymbol{e}_{\phi } \end{equation*}
    \end{document}

```python
check('linear_EM_waves')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} g = \left[\begin{array}{cccc}1 & \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  & \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  & 0\\\left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  & 1 & \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  & 0\\\left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  & \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  & 1 & 0\\0 & 0 & 0 & -1\end{array}\right] \end{equation*}
    \begin{equation*} X = x_{E} \boldsymbol{e}_{E} + x_{B} \boldsymbol{e}_{B} + x_{k} \boldsymbol{e}_{k} + t \boldsymbol{e}_{t} \end{equation*}
    \begin{equation*} K = k \boldsymbol{e}_{k} + \omega  \boldsymbol{e}_{t} \end{equation*}
    \begin{equation*} K\cdot X = \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  k x_{B} + \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  k x_{E} + k x_{k} - \omega  t \end{equation*}
    \begin{equation*} F = \frac{\left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  B e^{i \left(\left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  k x_{B} + \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  k x_{E} + k x_{k} - \omega  t\right)}}{\sqrt{- \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right ) ^{2} + 2 \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right ) ^{2} - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right ) ^{2} + 1}} \boldsymbol{e}_{E}\wedge \boldsymbol{e}_{B} - \frac{B e^{i \left(\left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  k x_{B} + \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  k x_{E} + k x_{k} - \omega  t\right)}}{\sqrt{- \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right ) ^{2} + 2 \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right ) ^{2} - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right ) ^{2} + 1}} \boldsymbol{e}_{E}\wedge \boldsymbol{e}_{k} + E e^{i \left(\left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  k x_{B} + \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  k x_{E} + k x_{k} - \omega  t\right)} \boldsymbol{e}_{E}\wedge \boldsymbol{e}_{t} + \frac{\left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  B e^{i \left(\left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  k x_{B} + \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  k x_{E} + k x_{k} - \omega  t\right)}}{\sqrt{- \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right ) ^{2} + 2 \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right ) ^{2} - \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right ) ^{2} + 1}} \boldsymbol{e}_{B}\wedge \boldsymbol{e}_{k} \end{equation*}
    \begin{equation*} \mbox{Substituting }e_{E}\cdot e_{B} = e_{E}\cdot e_{k} = e_{B}\cdot e_{k} = 0 \end{equation*}
    \begin{equation*} g = \left[\begin{array}{cccc}1 & \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  & \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  & 0\\\left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{B}\right )  & 1 & \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  & 0\\\left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  & \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  & 1 & 0\\0 & 0 & 0 & -1\end{array}\right] \end{equation*}
    \begin{equation*} X = x_{E} \boldsymbol{e}_{E} + x_{B} \boldsymbol{e}_{B} + x_{k} \boldsymbol{e}_{k} + t \boldsymbol{e}_{t} \end{equation*}
    \begin{equation*} K = k \boldsymbol{e}_{k} + \omega  \boldsymbol{e}_{t} \end{equation*}
    \begin{equation*} K\cdot X = \left (\boldsymbol{e}_{B}\cdot \boldsymbol{e}_{k}\right )  k x_{B} + \left (\boldsymbol{e}_{E}\cdot \boldsymbol{e}_{k}\right )  k x_{E} + k x_{k} - \omega  t \end{equation*}
    \begin{equation*} \mbox{Substituting }e_{E}\cdot e_{B} = e_{E}\cdot e_{k} = e_{B}\cdot e_{k} = 0 \end{equation*}
    \end{document}

```python
check('manifold')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} (u,v)\rightarrow (r,\theta,\phi) =  [1, u, v] \end{equation*}
    Unit Sphere Manifold:
    \begin{equation*} g = \left[\begin{array}{cc}1 & 0\\0 & {\sin{\left (u \right )}}^{2}\end{array}\right] \end{equation*}
    \begin{equation*} a = a^{u} \boldsymbol{e}_{u} + a^{v} \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} f = f^{u}  \boldsymbol{e}_{u} + f^{v}  \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} \nabla = \boldsymbol{e}_{u} \frac{\partial}{\partial u} + \boldsymbol{e}_{v} \frac{1}{{\sin{\left (u \right )}}^{2}} \frac{\partial}{\partial v} \end{equation*}
    \begin{equation*} a\cdot\nabla = a^{u} \frac{\partial}{\partial u} + a^{v} \frac{\partial}{\partial v} \end{equation*}
    \begin{equation*} \paren{a\cdot\nabla}\bm{e}_u = \frac{a^{v}}{\tan{\left (u \right )}} \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} \paren{a\cdot\nabla}\bm{e}_v = - \frac{a^{v} \sin{\left (2 u \right )}}{2} \boldsymbol{e}_{u} + \frac{a^{u}}{\tan{\left (u \right )}} \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} \paren{a\cdot\nabla}f = \left ( a^{u} \partial_{u} f^{u}  - \frac{a^{v} f^{v}  \sin{\left (2 u \right )}}{2} + a^{v} \partial_{v} f^{u} \right ) \boldsymbol{e}_{u} + \left ( \frac{a^{u} f^{v} }{\tan{\left (u \right )}} + a^{u} \partial_{u} f^{v}  + \frac{a^{v} f^{u} }{\tan{\left (u \right )}} + a^{v} \partial_{v} f^{v} \right ) \boldsymbol{e}_{v} \end{equation*}
    \begin{equation*} \nabla f = \left ( \frac{f^{u} }{\tan{\left (u \right )}} + \partial_{u} f^{u}  + \partial_{v} f^{v} \right )  + \left ( \frac{2 f^{v} }{\tan{\left (u \right )}} + \partial_{u} f^{v}  - \frac{\partial_{v} f^{u} }{{\sin{\left (u \right )}}^{2}}\right ) \boldsymbol{e}_{u}\wedge \boldsymbol{e}_{v} \end{equation*}
    1-D Manifold On Unit Sphere:
    \begin{equation*} \nabla = \boldsymbol{e}_{s} \frac{1}{{\sin{\left (u^{s}  \right )}}^{2} \left(\partial_{s} v^{s} \right)^{2} + \left(\partial_{s} u^{s} \right)^{2}} \frac{\partial}{\partial s} \end{equation*}
    \begin{equation*} \nabla g = \frac{\partial_{s} g }{{\sin{\left (u^{s}  \right )}}^{2} \left(\partial_{s} v^{s} \right)^{2} + \left(\partial_{s} u^{s} \right)^{2}} \boldsymbol{e}_{s} \end{equation*}
    \begin{equation*} \nabla \cdot \bm{h} = \frac{\left({\sin{\left (u^{s}  \right )}}^{2} \left(\partial_{s} v^{s} \right)^{2} + \left(\partial_{s} u^{s} \right)^{2}\right) \partial_{s} h^{s}  + \left({\sin{\left (u^{s}  \right )}}^{2} \partial_{s} v^{s}  \partial^{2}_{s} v^{s}  + \partial_{s} u^{s}  \partial^{2}_{s} u^{s} \right) h^{s}  + \frac{h^{s}  \sin{\left (2 u^{s}  \right )} \partial_{s} u^{s}  \left(\partial_{s} v^{s} \right)^{2}}{2}}{{\sin{\left (u^{s}  \right )}}^{2} \left(\partial_{s} v^{s} \right)^{2} + \left(\partial_{s} u^{s} \right)^{2}} \end{equation*}
    \end{document}

```python
check('matrix_latex')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
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    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
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    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} \left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right] \left[\begin{array}{c}5\\6\end{array}\right] = \left[\begin{array}{c}17\\39\end{array}\right] \end{equation*}
    \begin{equation*} \left[\begin{array}{cc}x^{3} & y^{3}\end{array}\right] \left[\begin{array}{cc}x^{2} & 2 x y\\2 x y & y^{2}\end{array}\right] = \left[\begin{array}{cc}x^{5} + 2 x y^{4} & 2 x^{4} y + y^{5}\end{array}\right] \end{equation*}
    \end{document}

```python
check('new_bug_grad_exp')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    Results with all scalar variables declared as real
    \begin{equation*} X = t \boldsymbol{\gamma }_{t} + x \boldsymbol{\gamma }_{x} + y \boldsymbol{\gamma }_{y} + z \boldsymbol{\gamma }_{z} + w \boldsymbol{\gamma }_{w} \end{equation*}
    \begin{equation*} K = k^{t} \boldsymbol{\gamma }_{t} + k^{x} \boldsymbol{\gamma }_{x} + k^{y} \boldsymbol{\gamma }_{y} + k^{z} \boldsymbol{\gamma }_{z} + k^{w} \boldsymbol{\gamma }_{w} \end{equation*}
    \begin{equation*} K\cdot X = k^{t} t - k^{w} w - k^{x} x - k^{y} y - k^{z} z \end{equation*}
    \begin{equation*} I^{2} = 1 \end{equation*}
    \begin{equation*} I_{xyzw} = I\gamma_{x}\gamma_{y}\gamma_{z}\gamma_{w} =  \boldsymbol{\gamma }_{t} \end{equation*}
    \begin{equation*} \lp I\gamma_{x}\gamma_{y}\gamma_{z}\gamma_{w}\rp^{2} = 1 \end{equation*}
    \begin{equation*} e^{I_{xyzw}K\cdot X} = \cosh{\left (- k^{t} t + k^{w} w + k^{x} x + k^{y} y + k^{z} z \right )}  - \sinh{\left (- k^{t} t + k^{w} w + k^{x} x + k^{y} y + k^{z} z \right )} \boldsymbol{\gamma }_{t} \end{equation*}
    \end{document}

```python
check('physics_check_latex')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
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    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\color{gray},frame=single]
    def General_Lorentz_Tranformation():
        Print_Function()
        (alpha,beta,gamma) = symbols('alpha beta gamma')
        (x,y,z,t) = symbols("x y z t",real=True)
        (st4d,g0,g1,g2,g3) = Ga.build('gamma*t|x|y|z',g=[1,-1,-1,-1])
        B = (x*g1+y*g2+z*g3)^(t*g0)
        print(B)
        print(B.exp(hint='+'))
        print(B.exp(hint='-'))
    \end{lstlisting}
    Code Output:
    \begin{equation*} - t x \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} - t y \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y} - t z \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} \cosh{\left (\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right| \right )}  - \frac{t x \sinh{\left (\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right| \right )}}{\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right|} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} - \frac{t y \sinh{\left (\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right| \right )}}{\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right|} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y} - \frac{t z \sinh{\left (\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right| \right )}}{\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right|} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} \cosh{\left (\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right| \right )}  - \frac{t x \sinh{\left (\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right| \right )}}{\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right|} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} - \frac{t y \sinh{\left (\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right| \right )}}{\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right|} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y} - \frac{t z \sinh{\left (\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right| \right )}}{\sqrt{x^{2} + y^{2} + z^{2}} \left|{t}\right|} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} \end{equation*}
    \end{document}

```python
check('print_check_latex')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} \bm{A} = A  + A^{x} \boldsymbol{e}_{x} + A^{y} \boldsymbol{e}_{y} + A^{z} \boldsymbol{e}_{z} + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + A^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + A^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} + A^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \bm{A} = A^{x}  \boldsymbol{e}_{x} + A^{y}  \boldsymbol{e}_{y} + A^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \bm{B} = B^{xy}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + B^{xz}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + B^{yz}  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \partial_{x} f  \boldsymbol{e}_{x} + \partial_{y} f  \boldsymbol{e}_{y} + \partial_{z} f  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot \bm{A} = \partial_{x} A^{x}  + \partial_{y} A^{y}  + \partial_{z} A^{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \bm{A} = \left ( \partial_{x} A^{x}  + \partial_{y} A^{y}  + \partial_{z} A^{z} \right )  + \left ( - \partial_{y} A^{x}  + \partial_{x} A^{y} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( - \partial_{z} A^{x}  + \partial_{x} A^{z} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( - \partial_{z} A^{y}  + \partial_{y} A^{z} \right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} -I (\boldsymbol{\nabla} \W \bm{A}) = \left ( - \partial_{z} A^{y}  + \partial_{y} A^{z} \right ) \boldsymbol{e}_{x} + \left ( \partial_{z} A^{x}  - \partial_{x} A^{z} \right ) \boldsymbol{e}_{y} + \left ( - \partial_{y} A^{x}  + \partial_{x} A^{y} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \bm{B} = \left ( - \partial_{y} B^{xy}  - \partial_{z} B^{xz} \right ) \boldsymbol{e}_{x} + \left ( \partial_{x} B^{xy}  - \partial_{z} B^{yz} \right ) \boldsymbol{e}_{y} + \left ( \partial_{x} B^{xz}  + \partial_{y} B^{yz} \right ) \boldsymbol{e}_{z} + \left ( \partial_{z} B^{xy}  - \partial_{y} B^{xz}  + \partial_{x} B^{yz} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \W \bm{B} = \left ( \partial_{z} B^{xy}  - \partial_{y} B^{xz}  + \partial_{x} B^{yz} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot \bm{B} = \left ( - \partial_{y} B^{xy}  - \partial_{z} B^{xz} \right ) \boldsymbol{e}_{x} + \left ( \partial_{x} B^{xy}  - \partial_{z} B^{yz} \right ) \boldsymbol{e}_{y} + \left ( \partial_{x} B^{xz}  + \partial_{y} B^{yz} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} g_{ij} = \left[\begin{array}{cccc}\left (\boldsymbol{a}\cdot \boldsymbol{a}\right )  & \left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  & \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{a}\cdot \boldsymbol{d}\right ) \\\left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{b}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{d}\right ) \\\left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{c}\cdot \boldsymbol{c}\right )  & \left (\boldsymbol{c}\cdot \boldsymbol{d}\right ) \\\left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{c}\cdot \boldsymbol{d}\right )  & \left (\boldsymbol{d}\cdot \boldsymbol{d}\right ) \end{array}\right] \end{equation*}
    \begin{equation*} \bm{a\cdot (b c)} = - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \boldsymbol{b} + \left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  \boldsymbol{c} \end{equation*}
    \begin{equation*} \bm{a\cdot (b\W c)} = - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \boldsymbol{b} + \left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  \boldsymbol{c} \end{equation*}
    \begin{equation*} \bm{a\cdot (b\W c\W d)} = \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \boldsymbol{b}\wedge \boldsymbol{c} - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \boldsymbol{b}\wedge \boldsymbol{d} + \left (\boldsymbol{a}\cdot \boldsymbol{b}\right )  \boldsymbol{c}\wedge \boldsymbol{d} \end{equation*}
    \begin{equation*} \bm{a\cdot (b\W c)+c\cdot (a\W b)+b\cdot (c\W a)} =  0 \end{equation*}
    \begin{equation*} \bm{a (b\W c)-b (a\W c)+c (a\W b)} = 3 \boldsymbol{a}\wedge \boldsymbol{b}\wedge \boldsymbol{c} \end{equation*}
    \begin{equation*} \bm{a (b\W c\W d)-b (a\W c\W d)+c (a\W b\W d)-d (a\W b\W c)} = 4 \boldsymbol{a}\wedge \boldsymbol{b}\wedge \boldsymbol{c}\wedge \boldsymbol{d} \end{equation*}
    \begin{equation*} \bm{(a\W b)\cdot (c\W d)} = - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  + \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{c}\right ) \end{equation*}
    \begin{equation*} \bm{((a\W b)\cdot c)\cdot d} = - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  + \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \left (\boldsymbol{b}\cdot \boldsymbol{c}\right ) \end{equation*}
    \begin{equation*} \bm{(a\W b)\times (c\W d)} = - \left (\boldsymbol{b}\cdot \boldsymbol{d}\right )  \boldsymbol{a}\wedge \boldsymbol{c} + \left (\boldsymbol{b}\cdot \boldsymbol{c}\right )  \boldsymbol{a}\wedge \boldsymbol{d} + \left (\boldsymbol{a}\cdot \boldsymbol{d}\right )  \boldsymbol{b}\wedge \boldsymbol{c} - \left (\boldsymbol{a}\cdot \boldsymbol{c}\right )  \boldsymbol{b}\wedge \boldsymbol{d} \end{equation*}
    \begin{equation*} E =  \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} E^{2} = \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right ) ^{2} - 2 \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right ) ^{2} + \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) ^{2} - 1 \end{equation*}
    \begin{equation*} E1 = (e2\W e3) E = \left ( \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) ^{2} - 1\right ) \boldsymbol{e}_{1} + \left ( \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{2} + \left ( - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} E2 =-(e1\W e3) E = \left ( \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{1} + \left ( \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right ) ^{2} - 1\right ) \boldsymbol{e}_{2} + \left ( - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} E3 = (e1\W e2) E = \left ( - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{1} + \left ( - \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right )  \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{3}\right )  + \left (\boldsymbol{e}_{2}\cdot \boldsymbol{e}_{3}\right ) \right ) \boldsymbol{e}_{2} + \left ( \left (\boldsymbol{e}_{1}\cdot \boldsymbol{e}_{2}\right ) ^{2} - 1\right ) \boldsymbol{e}_{3} \end{equation*}
    \begin{equation*} E1\cdot e2 =  0 \end{equation*}
    \begin{equation*} E1\cdot e3 =  0 \end{equation*}
    \begin{equation*} E2\cdot e1 =  0 \end{equation*}
    \begin{equation*} E2\cdot e3 =  0 \end{equation*}
    \begin{equation*} E3\cdot e1 =  0 \end{equation*}
    \begin{equation*} E3\cdot e2 =  0 \end{equation*}
    \begin{equation*} (E1\cdot e1)/E^{2} = 1 \end{equation*}
    \begin{equation*} (E2\cdot e2)/E^{2} = 1 \end{equation*}
    \begin{equation*} (E3\cdot e3)/E^{2} = 1 \end{equation*}
    \begin{equation*} A = A^{r}  \boldsymbol{e}_{r} + A^{\theta }  \boldsymbol{e}_{\theta } + A^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} B = B^{r\theta }  \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\theta } + B^{r\phi }  \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\phi } + B^{\theta \phi }  \boldsymbol{e}_{\theta }\wedge \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \partial_{r} f  \boldsymbol{e}_{r} + \frac{\partial_{\theta } f }{r} \boldsymbol{e}_{\theta } + \frac{\partial_{\phi } f }{r \sin{\left (\theta  \right )}} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \cdot A = \frac{r \partial_{r} A^{r}  + 2 A^{r}  + \frac{A^{\theta } }{\tan{\left (\theta  \right )}} + \partial_{\theta } A^{\theta }  + \frac{\partial_{\phi } A^{\phi } }{\sin{\left (\theta  \right )}}}{r} \end{equation*}
    \begin{equation*} -I (\boldsymbol{\nabla} \W A) = \frac{\frac{A^{\phi } }{\tan{\left (\theta  \right )}} + \partial_{\theta } A^{\phi }  - \frac{\partial_{\phi } A^{\theta } }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{r} + \frac{- r \partial_{r} A^{\phi }  - A^{\phi }  + \frac{\partial_{\phi } A^{r} }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{\theta } + \frac{r \partial_{r} A^{\theta }  + A^{\theta }  - \partial_{\theta } A^{r} }{r} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla} \W B = \frac{r \partial_{r} B^{\theta \phi }  - \frac{B^{r\phi } }{\tan{\left (\theta  \right )}} + 2 B^{\theta \phi }  - \partial_{\theta } B^{r\phi }  + \frac{\partial_{\phi } B^{r\theta } }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\theta }\wedge \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} B = \bm{B\gamma_{t}} = - B^{x}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} - B^{y}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y} - B^{z}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} E = \bm{E\gamma_{t}} = - E^{x}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} - E^{y}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y} - E^{z}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} F = E+IB = - E^{x}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} - E^{y}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y} - E^{z}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} - B^{z}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} + B^{y}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} - B^{x}  \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} J = J^{t}  \boldsymbol{\gamma }_{t} + J^{x}  \boldsymbol{\gamma }_{x} + J^{y}  \boldsymbol{\gamma }_{y} + J^{z}  \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  F = J \end{equation*}
    \begin{equation*} R = \cosh{\left (\frac{\alpha }{2} \right )}  + \sinh{\left (\frac{\alpha }{2} \right )} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} \end{equation*}
    \begin{equation*} t\bm{\gamma_{t}}+x\bm{\gamma_{x}} = t'\bm{\gamma'_{t}}+x'\bm{\gamma'_{x}} = R\lp t'\bm{\gamma_{t}}+x'\bm{\gamma_{x}}\rp R^{\dagger} \end{equation*}
    \begin{equation*} t\bm{\gamma_{t}}+x\bm{\gamma_{x}} = \left ( t' \cosh{\left (\alpha  \right )} - x' \sinh{\left (\alpha  \right )}\right ) \boldsymbol{\gamma }_{t} + \left ( - t' \sinh{\left (\alpha  \right )} + x' \cosh{\left (\alpha  \right )}\right ) \boldsymbol{\gamma }_{x} \end{equation*}
    \begin{equation*} \f{\sinh}{\alpha} = \gamma\beta \end{equation*}
    \begin{equation*} \f{\cosh}{\alpha} = \gamma \end{equation*}
    \begin{equation*} t\bm{\gamma_{t}}+x\bm{\gamma_{x}} = \gamma  \left(- \beta  x' + t'\right) \boldsymbol{\gamma }_{t} + \gamma  \left(- \beta  t' + x'\right) \boldsymbol{\gamma }_{x} \end{equation*}
    \begin{equation*} \bm{A} = A^{t}  \boldsymbol{\gamma }_{t} + A^{x}  \boldsymbol{\gamma }_{x} + A^{y}  \boldsymbol{\gamma }_{y} + A^{z}  \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} \bm{\psi} = \psi    + \psi ^{tx}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} + \psi ^{ty}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y} + \psi ^{tz}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} + \psi ^{xy}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} + \psi ^{xz}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + \psi ^{yz}  \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + \psi ^{txyz}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}
    \begin{equation*} value = 1 \end{equation*}
    \begin{align*} \text{untouched} \end{align*}
    \end{document}

```python
check('products_latex')
```

    \documentclass[10pt,fleqn]{report}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    3D Orthogonal Metric\newline
    Multvectors:
    \begin{equation*} s = s \end{equation*}
    \begin{equation*} v = v^{x} \boldsymbol{e}_{x} + v^{y} \boldsymbol{e}_{y} + v^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b = b^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    Products:
    \begin{equation*} s   s = s^{2} \end{equation*}
    \begin{equation*} s \W  s = s^{2} \end{equation*}
    \begin{equation*} s \rfloor  s = s^{2} \end{equation*}
    \begin{equation*} s \lfloor  s = s^{2} \end{equation*}
    \begin{equation*} s   v = s v^{x} \boldsymbol{e}_{x} + s v^{y} \boldsymbol{e}_{y} + s v^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \W  v = s v^{x} \boldsymbol{e}_{x} + s v^{y} \boldsymbol{e}_{y} + s v^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \rfloor  v = s v^{x} \boldsymbol{e}_{x} + s v^{y} \boldsymbol{e}_{y} + s v^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \lfloor  v =  0 \end{equation*}
    \begin{equation*} s   b = b^{xy} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz} s \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \W  b = b^{xy} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz} s \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \rfloor  b = b^{xy} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz} s \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \lfloor  b =  0 \end{equation*}
    \begin{equation*} v   s = s v^{x} \boldsymbol{e}_{x} + s v^{y} \boldsymbol{e}_{y} + s v^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \W  s = s v^{x} \boldsymbol{e}_{x} + s v^{y} \boldsymbol{e}_{y} + s v^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \rfloor  s =  0 \end{equation*}
    \begin{equation*} v \lfloor  s = s v^{x} \boldsymbol{e}_{x} + s v^{y} \boldsymbol{e}_{y} + s v^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v   v = {\left ( v^{x} \right )}^{2} + {\left ( v^{y} \right )}^{2} + {\left ( v^{z} \right )}^{2} \end{equation*}
    \begin{equation*} v \W  v =  0 \end{equation*}
    \begin{equation*} v \cdot  v = {\left ( v^{x} \right )}^{2} + {\left ( v^{y} \right )}^{2} + {\left ( v^{z} \right )}^{2} \end{equation*}
    \begin{equation*} v \rfloor  v = {\left ( v^{x} \right )}^{2} + {\left ( v^{y} \right )}^{2} + {\left ( v^{z} \right )}^{2} \end{equation*}
    \begin{equation*} v \lfloor  v = {\left ( v^{x} \right )}^{2} + {\left ( v^{y} \right )}^{2} + {\left ( v^{z} \right )}^{2} \end{equation*}
    \begin{equation*} v   b = \left ( - b^{xy} v^{y} - b^{xz} v^{z}\right ) \boldsymbol{e}_{x} + \left ( b^{xy} v^{x} - b^{yz} v^{z}\right ) \boldsymbol{e}_{y} + \left ( b^{xz} v^{x} + b^{yz} v^{y}\right ) \boldsymbol{e}_{z} + \left ( b^{xy} v^{z} - b^{xz} v^{y} + b^{yz} v^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \W  b = \left ( b^{xy} v^{z} - b^{xz} v^{y} + b^{yz} v^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \cdot  b = \left ( - b^{xy} v^{y} - b^{xz} v^{z}\right ) \boldsymbol{e}_{x} + \left ( b^{xy} v^{x} - b^{yz} v^{z}\right ) \boldsymbol{e}_{y} + \left ( b^{xz} v^{x} + b^{yz} v^{y}\right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \rfloor  b = \left ( - b^{xy} v^{y} - b^{xz} v^{z}\right ) \boldsymbol{e}_{x} + \left ( b^{xy} v^{x} - b^{yz} v^{z}\right ) \boldsymbol{e}_{y} + \left ( b^{xz} v^{x} + b^{yz} v^{y}\right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \lfloor  b =  0 \end{equation*}
    \begin{equation*} b   s = b^{xy} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz} s \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \W  s = b^{xy} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz} s \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \rfloor  s =  0 \end{equation*}
    \begin{equation*} b \lfloor  s = b^{xy} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz} s \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz} s \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b   v = \left ( b^{xy} v^{y} + b^{xz} v^{z}\right ) \boldsymbol{e}_{x} + \left ( - b^{xy} v^{x} + b^{yz} v^{z}\right ) \boldsymbol{e}_{y} + \left ( - b^{xz} v^{x} - b^{yz} v^{y}\right ) \boldsymbol{e}_{z} + \left ( b^{xy} v^{z} - b^{xz} v^{y} + b^{yz} v^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \W  v = \left ( b^{xy} v^{z} - b^{xz} v^{y} + b^{yz} v^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \cdot  v = \left ( b^{xy} v^{y} + b^{xz} v^{z}\right ) \boldsymbol{e}_{x} + \left ( - b^{xy} v^{x} + b^{yz} v^{z}\right ) \boldsymbol{e}_{y} + \left ( - b^{xz} v^{x} - b^{yz} v^{y}\right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \rfloor  v =  0 \end{equation*}
    \begin{equation*} b \lfloor  v = \left ( b^{xy} v^{y} + b^{xz} v^{z}\right ) \boldsymbol{e}_{x} + \left ( - b^{xy} v^{x} + b^{yz} v^{z}\right ) \boldsymbol{e}_{y} + \left ( - b^{xz} v^{x} - b^{yz} v^{y}\right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b   b = - {\left ( b^{xy} \right )}^{2} - {\left ( b^{xz} \right )}^{2} - {\left ( b^{yz} \right )}^{2} \end{equation*}
    \begin{equation*} b \W  b =  0 \end{equation*}
    \begin{equation*} b \cdot  b = - {\left ( b^{xy} \right )}^{2} - {\left ( b^{xz} \right )}^{2} - {\left ( b^{yz} \right )}^{2} \end{equation*}
    \begin{equation*} b \rfloor  b = - {\left ( b^{xy} \right )}^{2} - {\left ( b^{xz} \right )}^{2} - {\left ( b^{yz} \right )}^{2} \end{equation*}
    \begin{equation*} b \lfloor  b = - {\left ( b^{xy} \right )}^{2} - {\left ( b^{xz} \right )}^{2} - {\left ( b^{yz} \right )}^{2} \end{equation*}
    Multivector Functions:
    \begin{equation*} s(X) = s \end{equation*}
    \begin{equation*} v(X) = v^{x}  \boldsymbol{e}_{x} + v^{y}  \boldsymbol{e}_{y} + v^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b(X) = b^{xy}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz}  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    Products:
    \begin{equation*} \boldsymbol{\nabla}    s = \partial_{x} s  \boldsymbol{e}_{x} + \partial_{y} s  \boldsymbol{e}_{y} + \partial_{z} s  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \W  s = \partial_{x} s  \boldsymbol{e}_{x} + \partial_{y} s  \boldsymbol{e}_{y} + \partial_{z} s  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \rfloor  s =  0 \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \lfloor  s = \partial_{x} s  \boldsymbol{e}_{x} + \partial_{y} s  \boldsymbol{e}_{y} + \partial_{z} s  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}    v = \left ( \partial_{x} v^{x}  + \partial_{y} v^{y}  + \partial_{z} v^{z} \right )  + \left ( - \partial_{y} v^{x}  + \partial_{x} v^{y} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( - \partial_{z} v^{x}  + \partial_{x} v^{z} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( - \partial_{z} v^{y}  + \partial_{y} v^{z} \right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \W  v = \left ( - \partial_{y} v^{x}  + \partial_{x} v^{y} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( - \partial_{z} v^{x}  + \partial_{x} v^{z} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( - \partial_{z} v^{y}  + \partial_{y} v^{z} \right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \cdot  v = \partial_{x} v^{x}  + \partial_{y} v^{y}  + \partial_{z} v^{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \rfloor  v = \partial_{x} v^{x}  + \partial_{y} v^{y}  + \partial_{z} v^{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \lfloor  v = \partial_{x} v^{x}  + \partial_{y} v^{y}  + \partial_{z} v^{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}    b = \left ( - \partial_{y} b^{xy}  - \partial_{z} b^{xz} \right ) \boldsymbol{e}_{x} + \left ( \partial_{x} b^{xy}  - \partial_{z} b^{yz} \right ) \boldsymbol{e}_{y} + \left ( \partial_{x} b^{xz}  + \partial_{y} b^{yz} \right ) \boldsymbol{e}_{z} + \left ( \partial_{z} b^{xy}  - \partial_{y} b^{xz}  + \partial_{x} b^{yz} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \W  b = \left ( \partial_{z} b^{xy}  - \partial_{y} b^{xz}  + \partial_{x} b^{yz} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \cdot  b = \left ( - \partial_{y} b^{xy}  - \partial_{z} b^{xz} \right ) \boldsymbol{e}_{x} + \left ( \partial_{x} b^{xy}  - \partial_{z} b^{yz} \right ) \boldsymbol{e}_{y} + \left ( \partial_{x} b^{xz}  + \partial_{y} b^{yz} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \rfloor  b = \left ( - \partial_{y} b^{xy}  - \partial_{z} b^{xz} \right ) \boldsymbol{e}_{x} + \left ( \partial_{x} b^{xy}  - \partial_{z} b^{yz} \right ) \boldsymbol{e}_{y} + \left ( \partial_{x} b^{xz}  + \partial_{y} b^{yz} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \lfloor  b =  0 \end{equation*}
    \begin{equation*} s   \boldsymbol{\nabla}  = \boldsymbol{e}_{x} s  \frac{\partial}{\partial x} + \boldsymbol{e}_{y} s  \frac{\partial}{\partial y} + \boldsymbol{e}_{z} s  \frac{\partial}{\partial z} \end{equation*}
    \begin{equation*} s \W  \boldsymbol{\nabla}  = \boldsymbol{e}_{x} s  \frac{\partial}{\partial x} + \boldsymbol{e}_{y} s  \frac{\partial}{\partial y} + \boldsymbol{e}_{z} s  \frac{\partial}{\partial z} \end{equation*}
    \begin{equation*} s \rfloor  \boldsymbol{\nabla}  = \boldsymbol{e}_{x} s  \frac{\partial}{\partial x} + \boldsymbol{e}_{y} s  \frac{\partial}{\partial y} + \boldsymbol{e}_{z} s  \frac{\partial}{\partial z} \end{equation*}
    \begin{equation*} s \lfloor  \boldsymbol{\nabla}  =  0 \end{equation*}
    \begin{equation*} s   s = {s }^{2} \end{equation*}
    \begin{equation*} s \W  s = {s }^{2} \end{equation*}
    \begin{equation*} s \rfloor  s = {s }^{2} \end{equation*}
    \begin{equation*} s \lfloor  s = {s }^{2} \end{equation*}
    \begin{equation*} s   v = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} + s  v^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \W  v = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} + s  v^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \rfloor  v = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} + s  v^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \lfloor  v =  0 \end{equation*}
    \begin{equation*} s   b = b^{xy}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz}  s  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \W  b = b^{xy}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz}  s  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \rfloor  b = b^{xy}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz}  s  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} s \lfloor  b =  0 \end{equation*}
    \begin{equation*} v   \boldsymbol{\nabla}  = v^{x}  \frac{\partial}{\partial x} + v^{y}  \frac{\partial}{\partial y} + v^{z}  \frac{\partial}{\partial z} + \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \left ( - v^{y}  \frac{\partial}{\partial x} + v^{x}  \frac{\partial}{\partial y}\right )  + \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} \left ( - v^{z}  \frac{\partial}{\partial x} + v^{x}  \frac{\partial}{\partial z}\right )  + \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \left ( - v^{z}  \frac{\partial}{\partial y} + v^{y}  \frac{\partial}{\partial z}\right ) \end{equation*}
    \begin{equation*} v \W  \boldsymbol{\nabla}  = \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \left ( - v^{y}  \frac{\partial}{\partial x} + v^{x}  \frac{\partial}{\partial y}\right )  + \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} \left ( - v^{z}  \frac{\partial}{\partial x} + v^{x}  \frac{\partial}{\partial z}\right )  + \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \left ( - v^{z}  \frac{\partial}{\partial y} + v^{y}  \frac{\partial}{\partial z}\right ) \end{equation*}
    \begin{equation*} v \cdot  \boldsymbol{\nabla}  = v^{x}  \frac{\partial}{\partial x} + v^{y}  \frac{\partial}{\partial y} + v^{z}  \frac{\partial}{\partial z} \end{equation*}
    \begin{equation*} v \rfloor  \boldsymbol{\nabla}  = v^{x}  \frac{\partial}{\partial x} + v^{y}  \frac{\partial}{\partial y} + v^{z}  \frac{\partial}{\partial z} \end{equation*}
    \begin{equation*} v \lfloor  \boldsymbol{\nabla}  = v^{x}  \frac{\partial}{\partial x} + v^{y}  \frac{\partial}{\partial y} + v^{z}  \frac{\partial}{\partial z} \end{equation*}
    \begin{equation*} v   s = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} + s  v^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \W  s = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} + s  v^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \rfloor  s =  0 \end{equation*}
    \begin{equation*} v \lfloor  s = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} + s  v^{z}  \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v   v = {v^{x} }^{2} + {v^{y} }^{2} + {v^{z} }^{2} \end{equation*}
    \begin{equation*} v \W  v =  0 \end{equation*}
    \begin{equation*} v \cdot  v = {v^{x} }^{2} + {v^{y} }^{2} + {v^{z} }^{2} \end{equation*}
    \begin{equation*} v \rfloor  v = {v^{x} }^{2} + {v^{y} }^{2} + {v^{z} }^{2} \end{equation*}
    \begin{equation*} v \lfloor  v = {v^{x} }^{2} + {v^{y} }^{2} + {v^{z} }^{2} \end{equation*}
    \begin{equation*} v   b = \left ( - b^{xy}  v^{y}  - b^{xz}  v^{z} \right ) \boldsymbol{e}_{x} + \left ( b^{xy}  v^{x}  - b^{yz}  v^{z} \right ) \boldsymbol{e}_{y} + \left ( b^{xz}  v^{x}  + b^{yz}  v^{y} \right ) \boldsymbol{e}_{z} + \left ( b^{xy}  v^{z}  - b^{xz}  v^{y}  + b^{yz}  v^{x} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \W  b = \left ( b^{xy}  v^{z}  - b^{xz}  v^{y}  + b^{yz}  v^{x} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \cdot  b = \left ( - b^{xy}  v^{y}  - b^{xz}  v^{z} \right ) \boldsymbol{e}_{x} + \left ( b^{xy}  v^{x}  - b^{yz}  v^{z} \right ) \boldsymbol{e}_{y} + \left ( b^{xz}  v^{x}  + b^{yz}  v^{y} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \rfloor  b = \left ( - b^{xy}  v^{y}  - b^{xz}  v^{z} \right ) \boldsymbol{e}_{x} + \left ( b^{xy}  v^{x}  - b^{yz}  v^{z} \right ) \boldsymbol{e}_{y} + \left ( b^{xz}  v^{x}  + b^{yz}  v^{y} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} v \lfloor  b =  0 \end{equation*}
    \begin{equation*} b   \boldsymbol{\nabla}  = \boldsymbol{e}_{x} \left ( b^{xy}  \frac{\partial}{\partial y} + b^{xz}  \frac{\partial}{\partial z}\right )  + \boldsymbol{e}_{y} \left ( - b^{xy}  \frac{\partial}{\partial x} + b^{yz}  \frac{\partial}{\partial z}\right )  + \boldsymbol{e}_{z} \left ( - b^{xz}  \frac{\partial}{\partial x} -  b^{yz}  \frac{\partial}{\partial y}\right )  + \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \left ( b^{yz}  \frac{\partial}{\partial x} -  b^{xz}  \frac{\partial}{\partial y} + b^{xy}  \frac{\partial}{\partial z}\right ) \end{equation*}
    \begin{equation*} b \W  \boldsymbol{\nabla}  = \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \left ( b^{yz}  \frac{\partial}{\partial x} -  b^{xz}  \frac{\partial}{\partial y} + b^{xy}  \frac{\partial}{\partial z}\right ) \end{equation*}
    \begin{equation*} b \cdot  \boldsymbol{\nabla}  = \boldsymbol{e}_{x} \left ( b^{xy}  \frac{\partial}{\partial y} + b^{xz}  \frac{\partial}{\partial z}\right )  + \boldsymbol{e}_{y} \left ( - b^{xy}  \frac{\partial}{\partial x} + b^{yz}  \frac{\partial}{\partial z}\right )  + \boldsymbol{e}_{z} \left ( - b^{xz}  \frac{\partial}{\partial x} -  b^{yz}  \frac{\partial}{\partial y}\right ) \end{equation*}
    \begin{equation*} b \rfloor  \boldsymbol{\nabla}  =  0 \end{equation*}
    \begin{equation*} b \lfloor  \boldsymbol{\nabla}  = \boldsymbol{e}_{x} \left ( b^{xy}  \frac{\partial}{\partial y} + b^{xz}  \frac{\partial}{\partial z}\right )  + \boldsymbol{e}_{y} \left ( - b^{xy}  \frac{\partial}{\partial x} + b^{yz}  \frac{\partial}{\partial z}\right )  + \boldsymbol{e}_{z} \left ( - b^{xz}  \frac{\partial}{\partial x} -  b^{yz}  \frac{\partial}{\partial y}\right ) \end{equation*}
    \begin{equation*} b   s = b^{xy}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz}  s  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \W  s = b^{xy}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz}  s  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \rfloor  s =  0 \end{equation*}
    \begin{equation*} b \lfloor  s = b^{xy}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + b^{xz}  s  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + b^{yz}  s  \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b   v = \left ( b^{xy}  v^{y}  + b^{xz}  v^{z} \right ) \boldsymbol{e}_{x} + \left ( - b^{xy}  v^{x}  + b^{yz}  v^{z} \right ) \boldsymbol{e}_{y} + \left ( - b^{xz}  v^{x}  - b^{yz}  v^{y} \right ) \boldsymbol{e}_{z} + \left ( b^{xy}  v^{z}  - b^{xz}  v^{y}  + b^{yz}  v^{x} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \W  v = \left ( b^{xy}  v^{z}  - b^{xz}  v^{y}  + b^{yz}  v^{x} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \cdot  v = \left ( b^{xy}  v^{y}  + b^{xz}  v^{z} \right ) \boldsymbol{e}_{x} + \left ( - b^{xy}  v^{x}  + b^{yz}  v^{z} \right ) \boldsymbol{e}_{y} + \left ( - b^{xz}  v^{x}  - b^{yz}  v^{y} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b \rfloor  v =  0 \end{equation*}
    \begin{equation*} b \lfloor  v = \left ( b^{xy}  v^{y}  + b^{xz}  v^{z} \right ) \boldsymbol{e}_{x} + \left ( - b^{xy}  v^{x}  + b^{yz}  v^{z} \right ) \boldsymbol{e}_{y} + \left ( - b^{xz}  v^{x}  - b^{yz}  v^{y} \right ) \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} b   b = - {b^{xy} }^{2} - {b^{xz} }^{2} - {b^{yz} }^{2} \end{equation*}
    \begin{equation*} b \W  b =  0 \end{equation*}
    \begin{equation*} b \cdot  b = - {b^{xy} }^{2} - {b^{xz} }^{2} - {b^{yz} }^{2} \end{equation*}
    \begin{equation*} b \rfloor  b = - {b^{xy} }^{2} - {b^{xz} }^{2} - {b^{yz} }^{2} \end{equation*}
    \begin{equation*} b \lfloor  b = - {b^{xy} }^{2} - {b^{xz} }^{2} - {b^{yz} }^{2} \end{equation*}
    General 2D Metric\newline
    Multivector Functions:
    \begin{equation*} s(X) = s \end{equation*}
    \begin{equation*} v(X) = v^{x}  \boldsymbol{e}_{x} + v^{y}  \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} b(X) = v^{xy}  \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \end{equation*}
    Products:
    \begin{equation*} \boldsymbol{\nabla}    s = \frac{- \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{y} s  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \boldsymbol{e}_{x} + \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \partial_{y} s  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \W  s = \frac{- \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{y} s  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \boldsymbol{e}_{x} + \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \partial_{y} s  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \cdot  s = Not Allowed \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \rfloor  s =  0 \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \lfloor  s = \frac{- \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{y} s  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \boldsymbol{e}_{x} + \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \partial_{y} s  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}    v = \left ( \partial_{x} v^{x}  + \partial_{y} v^{y} \right )  + \frac{- \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \partial_{y} v^{x}  + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} v^{x}  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{y} v^{y}  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} v^{y} }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \W  v = \frac{- \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \partial_{y} v^{x}  + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} v^{x}  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  \partial_{y} v^{y}  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  \partial_{x} v^{y} }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \cdot  v = \partial_{x} v^{x}  + \partial_{y} v^{y} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \rfloor  v = \partial_{x} v^{x}  + \partial_{y} v^{y} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  \lfloor  v = \partial_{x} v^{x}  + \partial_{y} v^{y} \end{equation*}
    \begin{equation*} s   \boldsymbol{\nabla}  = \boldsymbol{e}_{x} \left ( \frac{\left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial x} -  \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial y}\right )  + \boldsymbol{e}_{y} \left ( - \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial x} + \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial y}\right ) \end{equation*}
    \begin{equation*} s \W  \boldsymbol{\nabla}  = \boldsymbol{e}_{x} \left ( \frac{\left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial x} -  \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial y}\right )  + \boldsymbol{e}_{y} \left ( - \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial x} + \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial y}\right ) \end{equation*}
    \begin{equation*} s \cdot  \boldsymbol{\nabla}  = Not Allowed \end{equation*}
    \begin{equation*} s \rfloor  \boldsymbol{\nabla}  = \boldsymbol{e}_{x} \left ( \frac{\left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial x} -  \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial y}\right )  + \boldsymbol{e}_{y} \left ( - \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial x} + \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  s }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial y}\right ) \end{equation*}
    \begin{equation*} s \lfloor  \boldsymbol{\nabla}  =  0 \end{equation*}
    \begin{equation*} s   s = {s }^{2} \end{equation*}
    \begin{equation*} s \W  s = {s }^{2} \end{equation*}
    \begin{equation*} s \cdot  s = Not Allowed \end{equation*}
    \begin{equation*} s \rfloor  s = {s }^{2} \end{equation*}
    \begin{equation*} s \lfloor  s = {s }^{2} \end{equation*}
    \begin{equation*} s   v = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} s \W  v = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} s \cdot  v = Not Allowed \end{equation*}
    \begin{equation*} s \rfloor  v = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} s \lfloor  v =  0 \end{equation*}
    \begin{equation*} v   \boldsymbol{\nabla}  = v^{x}  \frac{\partial}{\partial x} + v^{y}  \frac{\partial}{\partial y} + \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \left ( - \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  v^{x}  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  v^{y} }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial x} + \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  v^{x}  + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  v^{y} }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial y}\right ) \end{equation*}
    \begin{equation*} v \W  \boldsymbol{\nabla}  = \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \left ( - \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  v^{x}  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  v^{y} }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial x} + \frac{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  v^{x}  + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  v^{y} }{\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  - \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) ^{2}} \frac{\partial}{\partial y}\right ) \end{equation*}
    \begin{equation*} v \cdot  \boldsymbol{\nabla}  = v^{x}  \frac{\partial}{\partial x} + v^{y}  \frac{\partial}{\partial y} \end{equation*}
    \begin{equation*} v \rfloor  \boldsymbol{\nabla}  = v^{x}  \frac{\partial}{\partial x} + v^{y}  \frac{\partial}{\partial y} \end{equation*}
    \begin{equation*} v \lfloor  \boldsymbol{\nabla}  = v^{x}  \frac{\partial}{\partial x} + v^{y}  \frac{\partial}{\partial y} \end{equation*}
    \begin{equation*} v   s = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} v \W  s = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} v \cdot  s = Not Allowed \end{equation*}
    \begin{equation*} v \rfloor  s =  0 \end{equation*}
    \begin{equation*} v \lfloor  s = s  v^{x}  \boldsymbol{e}_{x} + s  v^{y}  \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} v   v = \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  {v^{x} }^{2} + 2 \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  v^{x}  v^{y}  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  {v^{y} }^{2} \end{equation*}
    \begin{equation*} v \W  v =  0 \end{equation*}
    \begin{equation*} v \cdot  v = \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  {v^{x} }^{2} + 2 \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  v^{x}  v^{y}  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  {v^{y} }^{2} \end{equation*}
    \begin{equation*} v \rfloor  v = \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  {v^{x} }^{2} + 2 \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  v^{x}  v^{y}  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  {v^{y} }^{2} \end{equation*}
    \begin{equation*} v \lfloor  v = \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  {v^{x} }^{2} + 2 \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  v^{x}  v^{y}  + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  {v^{y} }^{2} \end{equation*}
    \end{document}

```python
check('simple_check_latex')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={14in,11in},total={13in,10in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} g_{ij} = \left[\begin{array}{ccc}\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  & \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  & \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right ) \\\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  & \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  & \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right ) \\\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  & \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  & \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right ) \end{array}\right] \end{equation*}
    \begin{equation*} A = A  + A^{x} \boldsymbol{e}_{x} + A^{y} \boldsymbol{e}_{y} + A^{z} \boldsymbol{e}_{z} + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + A^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + A^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} + A^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} A =  \begin{aligned}[t]  & A  \\  &  + A^{x} \boldsymbol{e}_{x} + A^{y} \boldsymbol{e}_{y} + A^{z} \boldsymbol{e}_{z} \\  &  + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + A^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + A^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \\  &  + A^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}  \end{aligned} \end{equation*}
    \begin{equation*} A =  \begin{aligned}[t]  & A  \\  &  + A^{x} \boldsymbol{e}_{x} \\  &  + A^{y} \boldsymbol{e}_{y} \\  &  + A^{z} \boldsymbol{e}_{z} \\  &  + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \\  &  + A^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} \\  &  + A^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \\  &  + A^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}  \end{aligned} \end{equation*}
    \begin{equation*} A_{+} = A  + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + A^{xz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + A^{yz} \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} A_{-} = A^{x} \boldsymbol{e}_{x} + A^{y} \boldsymbol{e}_{y} + A^{z} \boldsymbol{e}_{z} + A^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} X = X^{x} \boldsymbol{e}_{x} + X^{y} \boldsymbol{e}_{y} + X^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} Y = Y^{x} \boldsymbol{e}_{x} + Y^{y} \boldsymbol{e}_{y} + Y^{z} \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} X Y =  \begin{aligned}[t]  & \left ( \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} Y^{y} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{z} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{y} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{z} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{y} + \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{z}\right )  \\  &  + \left ( X^{x} Y^{y} - X^{y} Y^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( X^{x} Y^{z} - X^{z} Y^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( X^{y} Y^{z} - X^{z} Y^{y}\right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z}  \end{aligned} \end{equation*}
    \begin{equation*} X\W Y = \left ( X^{x} Y^{y} - X^{y} Y^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( X^{x} Y^{z} - X^{z} Y^{x}\right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( X^{y} Y^{z} - X^{z} Y^{y}\right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}
    \begin{equation*} X\cdot Y = \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} Y^{y} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{x} Y^{z} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y} Y^{y} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{y} Y^{z} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{y} + \left (\boldsymbol{e}_{z}\cdot \boldsymbol{e}_{z}\right )  X^{z} Y^{z} \end{equation*}
    \begin{equation*} g_{ij} = \left[\begin{array}{cc}\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  & \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right ) \\\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  & \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right ) \end{array}\right] \end{equation*}
    \begin{equation*} X = X^{x} \boldsymbol{e}_{x} + X^{y} \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} A = A  + A^{xy} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} X\cdot A = - A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{x} + A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} X\rfloor A = - A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{x} + A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{y} \end{equation*}
    \begin{equation*} A\lfloor X = A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{x} + \left (\boldsymbol{e}_{y}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{x} - A^{xy} \left(\left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{x}\right )  X^{x} + \left (\boldsymbol{e}_{x}\cdot \boldsymbol{e}_{y}\right )  X^{y}\right) \boldsymbol{e}_{y} \end{equation*}
    \end{document}

```python
check('christoffel_symbols')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={9in,10in},total={8in,9in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \begin{equation*} \bm{\mbox{Base manifold (three dimensional)}} \end{equation*}
    \begin{equation*} \bm{\mbox{Metric tensor (cartesian coordinates - norm = False)}} \end{equation*}
    \begin{equation*} g = \left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right] \end{equation*}
    \begin{equation*} \ \end{equation*}
    \begin{equation*} \bm{\mbox{Two dimensioanal submanifold - Unit sphere}} \end{equation*}
    \begin{equation*} \text{Basis not normalised} \end{equation*}
    \begin{equation*} (\theta,\phi)\rightarrow (r,\theta,\phi) =  \left[ 1, \  \theta , \  \phi \right] \end{equation*}
    \begin{equation*} e_\theta \cdot  e_\theta =  1 \end{equation*}
    \begin{equation*} e_\phi \cdot  e_\phi =  {\sin{\left (\theta  \right )}}^{2} \end{equation*}
    \begin{equation*} g = \left[\begin{array}{cc}1 & 0\\0 & {\sin{\left (\theta  \right )}}^{2}\end{array}\right] \end{equation*}
    \begin{equation*} \text{g\_inv = } \left[\begin{array}{cc}1 & 0\\0 & \frac{1}{{\sin{\left (\theta  \right )}}^{2}}\end{array}\right] \end{equation*}
    \begin{equation*} \text{Christoffel symbols of the first kind: } \end{equation*}
    \begin{equation*} \Gamma_{1, \alpha, \beta} =  \left[\begin{matrix}0 & 0\\0 & - \frac{\sin{\left (2 \theta  \right )}}{2}\end{matrix}\right] \quad \Gamma_{2, \alpha, \beta} =  \left[\begin{matrix}0 & \frac{\sin{\left (2 \theta  \right )}}{2}\\\frac{\sin{\left (2 \theta  \right )}}{2} & 0\end{matrix}\right] \end{equation*}
    \begin{equation*} \text{Christoffel symbols of the second kind: } \end{equation*}
    \begin{equation*} \Gamma^{1}_{\phantom{1,}\alpha, \beta} =  \left[\begin{matrix}0 & 0\\0 & - \frac{\sin{\left (2 \theta  \right )}}{2}\end{matrix}\right] \quad \Gamma^{2}_{\phantom{2,}\alpha, \beta} =  \left[\begin{matrix}0 & \frac{1}{\tan{\left (\theta  \right )}}\\\frac{1}{\tan{\left (\theta  \right )}} & 0\end{matrix}\right] \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  =  \boldsymbol{e}_{\theta } \frac{\partial}{\partial \theta } + \boldsymbol{e}_{\phi } \frac{1}{{\sin{\left (\theta  \right )}}^{2}} \frac{\partial}{\partial \phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \partial_{\theta } f  \boldsymbol{e}_{\theta } + \frac{\partial_{\phi } f }{{\sin{\left (\theta  \right )}}^{2}} \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} F = F^{\theta }  \boldsymbol{e}_{\theta } + F^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  F =  \left ( \frac{F^{\theta } }{\tan{\left (\theta  \right )}} + \partial_{\phi } F^{\phi }  + \partial_{\theta } F^{\theta } \right )  + \left ( \frac{2 F^{\phi } }{\tan{\left (\theta  \right )}} + \partial_{\theta } F^{\phi }  - \frac{\partial_{\phi } F^{\theta } }{{\sin{\left (\theta  \right )}}^{2}}\right ) \boldsymbol{e}_{\theta }\wedge \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \ \end{equation*}
    \begin{equation*} \mbox{One dimensioanal submanifold} \end{equation*}
    \begin{equation*} \mbox{Basis not normalised} \end{equation*}
    \begin{equation*} (\phi)\rightarrow (\theta,\phi) =  \left[ \frac{\pi}{8}, \  \phi \right] \end{equation*}
    \begin{equation*} e_\phi \cdot  e_\phi =  \frac{1}{2} - \frac{\sqrt{2}}{4} \end{equation*}
    \begin{equation*} g =  \left[\begin{array}{c}\frac{1}{2} - \frac{\sqrt{2}}{4}\end{array}\right] \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  =  \boldsymbol{e}_{\phi } \left ( 2 \sqrt{2} + 4\right ) \frac{\partial}{\partial \phi } \end{equation*}
    \begin{equation*} \nabla h =  \left(2 \sqrt{2} + 4\right) \partial_{\phi } h  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} H = H^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}
    \begin{equation*} \nabla H =  \partial_{\phi } H^{\phi } \end{equation*}
    \begin{equation*} \ \end{equation*}
    \end{document}

```python
check('colored_christoffel_symbols')
```

    \documentclass[10pt,fleqn]{report}
    \usepackage[vcentering]{geometry}
    \geometry{papersize={9in,10in},total={8in,9in}}

    \pagestyle{empty}
    \usepackage[latin1]{inputenc}
    \usepackage{amsmath}
    \usepackage{amsfonts}
    \usepackage{amssymb}
    \usepackage{amsbsy}
    \usepackage{tensor}
    \usepackage{listings}
    \usepackage{color}
    \usepackage{xcolor}
    \usepackage{bm}
    \usepackage{breqn}
    \definecolor{gray}{rgb}{0.95,0.95,0.95}
    \setlength{\parindent}{0pt}
    \DeclareMathOperator{\Tr}{Tr}
    \DeclareMathOperator{\Adj}{Adj}
    \newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}
    \newcommand{\lp}{\left (}
    \newcommand{\rp}{\right )}
    \newcommand{\paren}[1]{\lp {#1} \rp}
    \newcommand{\half}{\frac{1}{2}}
    \newcommand{\llt}{\left <}
    \newcommand{\rgt}{\right >}
    \newcommand{\abs}[1]{\left |{#1}\right | }
    \newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}
    \newcommand{\lbrc}{\left \{}
    \newcommand{\rbrc}{\right \}}
    \newcommand{\W}{\wedge}
    \newcommand{\prm}[1]{{#1}'}
    \newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}
    \newcommand{\R}{\dagger}
    \newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}
    \newcommand{\grade}[1]{\left < {#1} \right >}
    \newcommand{\f}[2]{{#1}\lp{#2}\rp}
    \newcommand{\eval}[2]{\left . {#1} \right |_{#2}}
    \newcommand{\Nabla}{\boldsymbol{\nabla}}
    \newcommand{\eb}{\boldsymbol{e}}
    \usepackage{float}
    \floatstyle{plain} % optionally change the style of the new float
    \newfloat{Code}{H}{myc}
    \lstloadlanguages{Python}

    \begin{document}
    \definecolor{airforceblue}{rgb}{0.36, 0.54, 0.66}
    \definecolor{applegreen}{rgb}{0.55, 0.71, 0.0}
    \definecolor{atomictangerine}{rgb}{1.0, 0.6, 0.4}
    \begin{equation*} \bm{\mbox{Base manifold (three dimensional)}} \end{equation*}
    \begin{equation*} \bm{\mbox{Metric tensor (cartesian coordinates - norm = False)}} \end{equation*}
    \begin{equation*} g = \left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right] \end{equation*}
    \begin{equation*} \ \end{equation*}
    \begin{equation*} \bm{\mbox{Two dimensioanal submanifold - Unit sphere}} \end{equation*}
    \begin{equation*} \text{Basis not normalised} \end{equation*}
    \begin{equation*} (\theta,\phi)\rightarrow (r,\theta,\phi) =  \left[ 1, \  {\color{airforceblue}\theta}, \  {\color{applegreen}\phi}\right] \end{equation*}
    \begin{equation*} e_\theta \cdot  e_\theta =  1 \end{equation*}
    \begin{equation*} e_\phi \cdot  e_\phi =  {\sin{\left ({\color{airforceblue}\theta} \right )}}^{2} \end{equation*}
    \begin{equation*} g = \left[\begin{array}{cc}1 & 0\\0 & {\sin{\left ({\color{airforceblue}\theta} \right )}}^{2}\end{array}\right] \end{equation*}
    \begin{equation*} \text{g\_inv = } \left[\begin{array}{cc}1 & 0\\0 & \frac{1}{{\sin{\left ({\color{airforceblue}\theta} \right )}}^{2}}\end{array}\right] \end{equation*}
    \begin{equation*} \text{Christoffel symbols of the first kind: } \end{equation*}
    \begin{equation*} \Gamma_{1, \alpha, \beta} =  \left[\begin{matrix}0 & 0\\0 & - \frac{\sin{\left (2 {\color{airforceblue}\theta} \right )}}{2}\end{matrix}\right] \quad \Gamma_{2, \alpha, \beta} =  \left[\begin{matrix}0 & \frac{\sin{\left (2 {\color{airforceblue}\theta} \right )}}{2}\\\frac{\sin{\left (2 {\color{airforceblue}\theta} \right )}}{2} & 0\end{matrix}\right] \end{equation*}
    \begin{equation*} \text{Christoffel symbols of the second kind: } \end{equation*}
    \begin{equation*} \Gamma^{1}_{\phantom{1,}\alpha, \beta} =  \left[\begin{matrix}0 & 0\\0 & - \frac{\sin{\left (2 {\color{airforceblue}\theta} \right )}}{2}\end{matrix}\right] \quad \Gamma^{2}_{\phantom{2,}\alpha, \beta} =  \left[\begin{matrix}0 & \frac{1}{\tan{\left ({\color{airforceblue}\theta} \right )}}\\\frac{1}{\tan{\left ({\color{airforceblue}\theta} \right )}} & 0\end{matrix}\right] \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  =  \boldsymbol{e}_{{\color{airforceblue}\theta}} \frac{\partial}{\partial {\color{airforceblue}\theta}} + \boldsymbol{e}_{{\color{applegreen}\phi}} \frac{1}{{\sin{\left ({\color{airforceblue}\theta} \right )}}^{2}} \frac{\partial}{\partial {\color{applegreen}\phi}} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  f = \partial_{{\color{airforceblue}\theta}} f  \boldsymbol{e}_{{\color{airforceblue}\theta}} + \frac{\partial_{{\color{applegreen}\phi}} f }{{\sin{\left ({\color{airforceblue}\theta} \right )}}^{2}} \boldsymbol{e}_{{\color{applegreen}\phi}} \end{equation*}
    \begin{equation*} F = F^{{\color{airforceblue}\theta}}  \boldsymbol{e}_{{\color{airforceblue}\theta}} + F^{{\color{applegreen}\phi}}  \boldsymbol{e}_{{\color{applegreen}\phi}} \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  F =  \left ( \frac{F^{{\color{airforceblue}\theta}} }{\tan{\left ({\color{airforceblue}\theta} \right )}} + \partial_{{\color{airforceblue}\theta}} F^{{\color{airforceblue}\theta}}  + \partial_{{\color{applegreen}\phi}} F^{{\color{applegreen}\phi}} \right )  + \left ( \frac{2 F^{{\color{applegreen}\phi}} }{\tan{\left ({\color{airforceblue}\theta} \right )}} + \partial_{{\color{airforceblue}\theta}} F^{{\color{applegreen}\phi}}  - \frac{\partial_{{\color{applegreen}\phi}} F^{{\color{airforceblue}\theta}} }{{\sin{\left ({\color{airforceblue}\theta} \right )}}^{2}}\right ) \boldsymbol{e}_{{\color{airforceblue}\theta}}\wedge \boldsymbol{e}_{{\color{applegreen}\phi}} \end{equation*}
    \begin{equation*} \ \end{equation*}
    \begin{equation*} \mbox{One dimensioanal submanifold} \end{equation*}
    \begin{equation*} \mbox{Basis not normalised} \end{equation*}
    \begin{equation*} (\phi)\rightarrow (\theta,\phi) =  \left[ \frac{\pi}{8}, \  {\color{atomictangerine}\phi}\right] \end{equation*}
    \begin{equation*} e_\phi \cdot  e_\phi =  \frac{1}{2} - \frac{\sqrt{2}}{4} \end{equation*}
    \begin{equation*} g =  \left[\begin{array}{c}\frac{1}{2} - \frac{\sqrt{2}}{4}\end{array}\right] \end{equation*}
    \begin{equation*} \boldsymbol{\nabla}  =  \boldsymbol{e}_{{\color{atomictangerine}\phi}} \left ( 2 \sqrt{2} + 4\right ) \frac{\partial}{\partial {\color{atomictangerine}\phi}} \end{equation*}
    \begin{equation*} \nabla h =  \left(2 \sqrt{2} + 4\right) \partial_{{\color{atomictangerine}\phi}} h  \boldsymbol{e}_{{\color{atomictangerine}\phi}} \end{equation*}
    \begin{equation*} H = H^{{\color{atomictangerine}\phi}}  \boldsymbol{e}_{{\color{atomictangerine}\phi}} \end{equation*}
    \begin{equation*} \nabla H =  \partial_{{\color{atomictangerine}\phi}} H^{{\color{atomictangerine}\phi}} \end{equation*}
    \begin{equation*} \ \end{equation*}
    \end{document}