In [1]:

    

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

Format()
coords = (et,ex,ey,ez) = symbols('t,x,y,z',real=True)
base=Ga('e*t|x|y|z',g=[1,-1,-1,-1],coords=symbols('t,x,y,z',real=True),wedge=False)

potential=base.mv('phi','vector',f=True)
potential


    

    
Out[1]:





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

In [2]:

    

field=base.grad*potential
field


    

    
Out[2]:





\begin{equation*} \left ( \partial_{t} \phi ^{t}  + \partial_{x} \phi ^{x}  + \partial_{y} \phi ^{y}  + \partial_{z} \phi ^{z} \right )  + \left ( \partial_{x} \phi ^{t}  + \partial_{t} \phi ^{x} \right ) \boldsymbol{e}_{tx} + \left ( \partial_{y} \phi ^{t}  + \partial_{t} \phi ^{y} \right ) \boldsymbol{e}_{ty} + \left ( \partial_{z} \phi ^{t}  + \partial_{t} \phi ^{z} \right ) \boldsymbol{e}_{tz} + \left ( \partial_{y} \phi ^{x}  - \partial_{x} \phi ^{y} \right ) \boldsymbol{e}_{xy} + \left ( \partial_{z} \phi ^{x}  - \partial_{x} \phi ^{z} \right ) \boldsymbol{e}_{xz} + \left ( \partial_{z} \phi ^{y}  - \partial_{y} \phi ^{z} \right ) \boldsymbol{e}_{yz} \end{equation*}

In [3]:

    

grad_field = base.grad*field
grad_field


    

    
Out[3]:





\begin{equation*} \left ( \partial^{2}_{t} \phi ^{t}  - \partial^{2}_{x} \phi ^{t}  - \partial^{2}_{y} \phi ^{t}  - \partial^{2}_{z} \phi ^{t} \right ) \boldsymbol{e}_{t} + \left ( \partial^{2}_{t} \phi ^{x}  - \partial^{2}_{x} \phi ^{x}  - \partial^{2}_{y} \phi ^{x}  - \partial^{2}_{z} \phi ^{x} \right ) \boldsymbol{e}_{x} + \left ( \partial^{2}_{t} \phi ^{y}  - \partial^{2}_{x} \phi ^{y}  - \partial^{2}_{y} \phi ^{y}  - \partial^{2}_{z} \phi ^{y} \right ) \boldsymbol{e}_{y} + \left ( \partial^{2}_{t} \phi ^{z}  - \partial^{2}_{x} \phi ^{z}  - \partial^{2}_{y} \phi ^{z}  - \partial^{2}_{z} \phi ^{z} \right ) \boldsymbol{e}_{z} \end{equation*}

In [4]:

    

part=field.proj([base.mv()[0]^base.mv()[1]])
part


    

    
Out[4]:





\begin{equation*} \left ( \partial_{x} \phi ^{t}  + \partial_{t} \phi ^{x} \right ) \boldsymbol{e}_{tx} \end{equation*}

In [5]:

    

dpart = base.grad*part
dpart


    

    
Out[5]:





\begin{equation*} \left ( - \partial^{2}_{x} \phi ^{t}  - \partial_{t}\partial_{x} \phi ^{x} \right ) \boldsymbol{e}_{t} + \left ( \partial^{2}_{t} \phi ^{x}  + \partial_{t}\partial_{x} \phi ^{t} \right ) \boldsymbol{e}_{x} + \left ( - \partial_{x}\partial_{y} \phi ^{t}  - \partial_{t}\partial_{y} \phi ^{x} \right ) \boldsymbol{e}_{txy} + \left ( - \partial_{x}\partial_{z} \phi ^{t}  - \partial_{t}\partial_{z} \phi ^{x} \right ) \boldsymbol{e}_{txz} \end{equation*}