In [1]:

    

from sympy import symbols, sin, cos, Abs
from galgebra.ga import Ga
from galgebra.printer import Format, Fmt
from IPython.display import Latex
Format()


    

In [2]:

    

xyz_coords = (x, y, z) = symbols('x y z', real=True)
(o3d, ex, ey, ez) = Ga.build('e', g=[1, 1, 1], coords=xyz_coords, norm=True)
o3d.g


    

    
Out[2]:





$\displaystyle \left[\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{matrix}\right]$

In [3]:

    

f = o3d.mv('f', 'scalar', f=True)
f


    

    
Out[3]:





\begin{equation*} f = f  \end{equation*}

In [4]:

    

F = o3d.mv('F', 'vector', f=True)
lap = o3d.grad*o3d.grad
lap.Fmt(1,r'\nabla^{2}')


    

    
Out[4]:





\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 [5]:

    

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


    

    
Out[5]:





\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 [6]:

    

lapf = lap*f
lapf


    

    
Out[6]:





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

In [7]:

    

lapf = o3d.grad | (o3d.grad * f)
lapf.Fmt(1,r'\nabla \cdot (\nabla f)')


    

    
Out[7]:





\begin{equation*} \nabla \cdot (\nabla f) = \partial^{2}_{x} f  + \partial^{2}_{y} f  + \partial^{2}_{z} f  \end{equation*}

In [8]:

    

divF = o3d.grad|F
divF.Fmt(1,'x =')


    

    
Out[8]:





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

In [9]:

    

gradF = o3d.grad * F
gradF.Fmt(1,r'\nabla F')


    

    
Out[9]:





\begin{equation*} \nabla F = \left ( \partial_{x} F^{x}  + \partial_{y} F^{y}  + \partial_{z} F^{z} \right )  + \left ( - \partial_{y} F^{x}  + \partial_{x} F^{y} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} + \left ( - \partial_{z} F^{x}  + \partial_{x} F^{z} \right ) \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} + \left ( - \partial_{z} F^{y}  + \partial_{y} F^{z} \right ) \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{equation*}

In [10]:

    

sph_coords = (r, th, phi) = symbols('r theta phi', real=True)
(sp3d, er, eth, ephi) = Ga.build('e', g=[1, r**2, r**2 * sin(th)**2], coords=sph_coords, norm=True)
sp3d.g_raw


    

    
Out[10]:





$\displaystyle \left[\begin{matrix}1 & 0 & 0\\0 & r^{2} & 0\\0 & 0 & r^{2} \sin^{2}{\left(\theta \right)}\end{matrix}\right]$

In [11]:

    

sp3d.grad.Fmt(1,r'\nabla')


    

    
Out[11]:





\begin{equation*} \nabla = \boldsymbol{e}_{r} \frac{\partial}{\partial r} + \boldsymbol{e}_{\theta } \frac{1}{r} \frac{\partial}{\partial \theta } + \boldsymbol{e}_{\phi } \frac{1}{r \sin{\left (\theta  \right )}} \frac{\partial}{\partial \phi } \end{equation*}

In [12]:

    

f = sp3d.mv('f', 'scalar', f=True)
F = sp3d.mv('F', 'vector', f=True)
B = sp3d.mv('B', 'bivector', f=True)
sp3d.grad.Fmt(1,r'\nabla')
lap = sp3d.grad*sp3d.grad
lap.Fmt(1,r'\nabla^{2} ')


    

    
Out[12]:





\begin{equation*} \nabla^{2}  = \frac{1}{r^{2} \tan{\left (\theta  \right )}} \frac{\partial}{\partial \theta } + \frac{2}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2} {\sin{\left (\theta  \right )}}^{2}} \frac{\partial^{2}}{\partial \phi ^{2}} + r^{-2} \frac{\partial^{2}}{\partial \theta ^{2}} + \frac{\partial^{2}}{\partial r^{2}} \end{equation*}

In [13]:

    

Lapf = lap*f
Lapf.Fmt(1,r'\nabla^{2} f')


    

    
Out[13]:





\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*}

In [14]:

    

lapf = sp3d.grad | (sp3d.grad * f)
lapf.Fmt(1,r'\nabla \cdot (\nabla f)')


    

    
Out[14]:





\begin{equation*} \nabla \cdot (\nabla 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*}

In [15]:

    

dviF = sp3d.grad | F
divF.Fmt(1,r'\nabla F')


    

    
Out[15]:





\begin{equation*} \nabla F = \partial_{x} F^{x}  + \partial_{y} F^{y}  + \partial_{z} F^{z}  \end{equation*}

In [16]:

    

curlF = sp3d.grad ^ F
curlF.Fmt(1,r'\nabla \wedge F')


    

    
Out[16]:





\begin{equation*} \nabla \wedge F = \frac{r \partial_{r} F^{\theta }  + F^{\theta }  - \partial_{\theta } F^{r} }{r} \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\theta } + \frac{r \partial_{r} F^{\phi }  + F^{\phi }  - \frac{\partial_{\phi } F^{r} }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{r}\wedge \boldsymbol{e}_{\phi } + \frac{\frac{F^{\phi } }{\tan{\left (\theta  \right )}} + \partial_{\theta } F^{\phi }  - \frac{\partial_{\phi } F^{\theta } }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{\theta }\wedge \boldsymbol{e}_{\phi } \end{equation*}

In [17]:

    

divB = sp3d.grad | B
divB.Fmt(1,r'\nabla \cdot B')


    

    
Out[17]:





\begin{equation*} \nabla \cdot B = - \frac{\frac{B^{r\theta } }{\tan{\left (\theta  \right )}} + \partial_{\theta } B^{r\theta }  + \frac{\partial_{\phi } B^{r\phi } }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{r} + \frac{r \partial_{r} B^{r\theta }  + B^{r\theta }  - \frac{\partial_{\phi } B^{\theta \phi } }{\sin{\left (\theta  \right )}}}{r} \boldsymbol{e}_{\theta } + \frac{r \partial_{r} B^{r\phi }  + B^{r\phi }  + \partial_{\theta } B^{\theta \phi } }{r} \boldsymbol{e}_{\phi } \end{equation*}

In [18]:

    

F


    

    
Out[18]:





\begin{equation*} F = F^{r}  \boldsymbol{e}_{r} + F^{\theta }  \boldsymbol{e}_{\theta } + F^{\phi }  \boldsymbol{e}_{\phi } \end{equation*}

In [19]:

    

F.Fmt(3,'F')


    

    
Out[19]:





 \begin{align*}   F =& F^{r}  \boldsymbol{e}_{r} \\  &  + F^{\theta }  \boldsymbol{e}_{\theta } \\  &  + F^{\phi }  \boldsymbol{e}_{\phi }  \end{align*} 

In [20]:

    

F.norm()


    

    
Out[20]:





$\displaystyle \sqrt{\operatorname{F^{\phi}}^{2}{\left(r,\theta,\phi \right)} + \operatorname{F^{r}}^{2}{\left(r,\theta,\phi \right)} + \operatorname{F^{\theta}}^{2}{\left(r,\theta,\phi \right)}}$