In [1]:

    

from galgebra.printer import Format
from galgebra.ga import Ga
from sympy import symbols


    

In [2]:

    

Format()


    

In [3]:

    

coords = (x,y,z) = symbols('x,y,z',real=True)
(o3d,ex,ey,ez) = Ga.build('e_x e_y e_z',g=[1,1,1],coords=coords)


    

In [4]:

    

v = o3d.mv('v','vector')


    

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v


    

    
Out[5]:





\begin{equation*} v = v^{x} \boldsymbol{e}_{x} + v^{y} \boldsymbol{e}_{y} + v^{z} \boldsymbol{e}_{z} \end{equation*}

In [6]:

    

v.Fmt(3,'v')


    

    
Out[6]:





 \begin{align*}   v =& v^{x} \boldsymbol{e}_{x} \\  &  + v^{y} \boldsymbol{e}_{y} \\  &  + v^{z} \boldsymbol{e}_{z}  \end{align*} 

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V = o3d.mv('V','vector',f=True)


    

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V.Fmt()


    

    
Out[8]:





\begin{equation*} V = V^{x}  \boldsymbol{e}_{x} + V^{y}  \boldsymbol{e}_{y} + V^{z}  \boldsymbol{e}_{z} \end{equation*}

In [9]:

    

V


    

    
Out[9]:





\begin{equation*} V = V^{x}  \boldsymbol{e}_{x} + V^{y}  \boldsymbol{e}_{y} + V^{z}  \boldsymbol{e}_{z} \end{equation*}

In [10]:

    

gradV = o3d.grad*V


    

In [11]:

    

gradV


    

    
Out[11]:





\begin{equation*} \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*}

In [12]:

    

gradV.Fmt(3,r'\nabla V')


    

    
Out[12]:





 \begin{align*}   \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{align*} 

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gradV.Fmt(2,r'\nabla V')


    

    
Out[13]:





 \begin{align*}   \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{align*} 

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lap = o3d.grad|o3d.grad


    

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lap


    

    
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\begin{equation*} \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}} \end{equation*}

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lap.Fmt(1,'\\nabla^{2}')


    

    
Out[16]:





\begin{equation*} \nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}} \end{equation*}

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A = o3d.lt('A')


    

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A


    

    
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\begin{equation*} \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*}

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A.Fmt(1,'A')


    

    
Out[19]:





\begin{equation*} \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*}

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A.Fmt(2,'A')


    

    
Out[20]:





\begin{equation*} \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*}

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