In [1]:

    

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


    

In [2]:

    

from sympy import symbols, latex, init_printing
Format(ipy=True)
init_printing()


    

In [3]:

    

coords = (x, y, z) = symbols('x y z', real=True)


    

In [4]:

    

o3d = Ga('e_x e_y e_z',g=[1,1,1], coords=coords)


    

In [5]:

    

o3d.grad


    

    
Out[5]:





$$ e_{x} \frac{\partial}{\partial x} + e_{y} \frac{\partial}{\partial y} + e_{z} \frac{\partial}{\partial z} $$

In [6]:

    

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


    

In [7]:

    

F


    

    
Out[7]:





\begin{equation*} F^{x}  e_{x} + F^{y}  e_{y} + F^{z}  e_{z} \end{equation*}

In [8]:

    

o3d.grad * F


    

    
Out[8]:





\begin{equation*} \left ( \partial_{x} F^{x}  + \partial_{y} F^{y}  + \partial_{z} F^{z} \right )  + \left ( - \partial_{y} F^{x}  + \partial_{x} F^{y} \right ) e_{x}\wedge e_{y} + \left ( - \partial_{z} F^{x}  + \partial_{x} F^{z} \right ) e_{x}\wedge e_{z} + \left ( - \partial_{z} F^{y}  + \partial_{y} F^{z} \right ) e_{y}\wedge e_{z} \end{equation*}

In [9]:

    

o3d.grad * o3d.grad


    

    
Out[9]:





$$ \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}} $$

In [10]:

    

(o3d.grad * o3d.grad) * F


    

    
Out[10]:





\begin{equation*} \left ( \partial^{2}_{x} F^{x}  + \partial^{2}_{y} F^{x}  + \partial^{2}_{z} F^{x} \right ) e_{x} + \left ( \partial^{2}_{x} F^{y}  + \partial^{2}_{y} F^{y}  + \partial^{2}_{z} F^{y} \right ) e_{y} + \left ( \partial^{2}_{x} F^{z}  + \partial^{2}_{y} F^{z}  + \partial^{2}_{z} F^{z} \right ) e_{z} \end{equation*}

In [11]:

    

ex, ey, ez = o3d.mv()
X = (x*ex)+(y*ey)+(z*ez)


    

In [12]:

    

X


    

    
Out[12]:





\begin{equation*} x e_{x} + y e_{y} + z e_{z} \end{equation*}

In [13]:

    

L = (X ^ o3d.grad) - (o3d.grad ^ X)


    

In [14]:

    

L


    

    
Out[14]:





$$ e_{x}\wedge e_{y} \left ( - y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}\right )  + e_{x}\wedge e_{z} \left ( - z \frac{\partial}{\partial x} + x \frac{\partial}{\partial z}\right )  + e_{y}\wedge e_{z} \left ( - z \frac{\partial}{\partial y} + y \frac{\partial}{\partial z}\right )  $$

In [15]:

    

Llst = (Lxy,Lyz,Lxz) = L.components()


    

In [16]:

    

Llst


    

    
Out[16]:





$$\begin{pmatrix}e_{x}\wedge e_{y} \left ( - y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}\right ) , & e_{x}\wedge e_{z} \left ( - z \frac{\partial}{\partial x} + x \frac{\partial}{\partial z}\right ) , & e_{y}\wedge e_{z} \left ( - z \frac{\partial}{\partial y} + y \frac{\partial}{\partial z}\right ) \end{pmatrix}$$

In [17]:

    

def Acom(x,y): return x*y+y*x #anticommutator since bases are included, not just components


    

In [18]:

    

Acom(Lxy,Lyz)


    

    
Out[18]:





$$ e_{y}\wedge e_{z} \left ( - z \frac{\partial}{\partial y} + y \frac{\partial}{\partial z}\right )  $$

In [19]:

    

Acom(Lyz,Lxz)


    

    
Out[19]:





$$ e_{x}\wedge e_{y} \left ( - y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}\right )  $$

In [20]:

    

Acom(Lxz,Lxy)


    

    
Out[20]:





$$ e_{x}\wedge e_{z} \left ( - z \frac{\partial}{\partial x} + x \frac{\partial}{\partial z}\right )  $$

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