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')

In [5]:

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{equation*} v = \begin{aligned}[t] & v^{x} \boldsymbol{e}_{x} \\ & + v^{y} \boldsymbol{e}_{y} \\ & + v^{z} \boldsymbol{e}_{z} \end{aligned} \end{equation*}

In [7]:

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

In [8]:

V.Fmt()

Out[8]:

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

In [9]:

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{equation*} \nabla V = \begin{aligned}[t] & \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{aligned} \end{equation*}

In [13]:

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

Out[13]:

\begin{equation*} \nabla V = \begin{aligned}[t] & \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{aligned} \end{equation*}

In [14]:

lap = o3d.grad|o3d.grad

In [15]:

lap

Out[15]:

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

In [16]:

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*}

In [17]:

A = o3d.lt('A')

In [18]:

Out[18]:

\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*}

In [19]:

A.Fmt(1,'A')

Out[19]:

\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*}

In [20]:

A.Fmt(2,'A')

Out[20]:

\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*}

In [ ]: