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# sp3: Spherical Coordinates in R^3

# Make SymPy available to this program:
import sympy
from sympy import *

# Make GAlgebra available to this program:
from galgebra.ga import *
from galgebra.mv import *
from galgebra.printer import Fmt, GaPrinter, Format
    # Fmt:       sets the way that a multivector's basis expansion is output.
    # GaPrinter: makes GA output a little more readable.
    # Format:    turns on latex printer.
from galgebra.gprinter import gFormat, gprint
gFormat()

$\displaystyle \DeclareMathOperator{\Tr}{Tr}$$ $$\DeclareMathOperator{\Adj}{Adj}$$ $$\newcommand{\bfrac}[2]{\displaystyle\frac{#1}{#2}}$$ $$\newcommand{\lp}{\left (}$$ $$\newcommand{\rp}{\right )}$$ $$\newcommand{\paren}[1]{\lp {#1} \rp}$$ $$\newcommand{\half}{\frac{1}{2}}$$ $$\newcommand{\llt}{\left <}$$ $$\newcommand{\rgt}{\right >}$$ $$\newcommand{\abs}[1]{\left |{#1}\right | }$$ $$\newcommand{\pdiff}[2]{\bfrac{\partial {#1}}{\partial {#2}}}$$ $$\newcommand{\npdiff}[3]{\bfrac{\partial^{#3} {#1}}{\partial {#2}^{#3}}}$$ $$\newcommand{\lbrc}{\left \{}$$ $$\newcommand{\rbrc}{\right \}}$$ $$\newcommand{\W}{\wedge}$$ $$\newcommand{\prm}[1]{{#1}^{\prime}}$$ $$\newcommand{\ddt}[1]{\bfrac{d{#1}}{dt}}$$ $$\newcommand{\R}{\dagger}$$ $$\newcommand{\deriv}[3]{\bfrac{d^{#3}#1}{d{#2}^{#3}}}$$ $$\newcommand{\grade}[2]{\left < {#1} \right >_{#2}}$$ $$\newcommand{\f}[2]{{#1}\lp {#2} \rp}$$ $$\newcommand{\eval}[2]{\left . {#1} \right |_{#2}}$$ $$\newcommand{\bs}[1]{\boldsymbol{#1}}$$ $$\newcommand{\grad}{\bs{\nabla}}$

# sp3: Geometric algebra for R^3 using spherical coordinates
# Mathematics convention: r, phi (colatitude), theta (longitude), in right handed order.

sp3coords = (r, phi, theta) = symbols('r phi theta')
sp3 = Ga('e', g=None, coords=sp3coords, \
    X=[r*sin(phi)*cos(theta), r*sin(phi)*sin(theta),r*cos(phi)], norm=True)
       # g = None. Instead, the spherical coordinate parameterization X is used.
       # "\" is Python's line continuation character
(er, ephi, etheta) = sp3.mv()

grad = sp3.grad
from galgebra.dop import *
pdr     = Pdop(r)
pdphi   = Pdop(phi)
pdtheta = Pdop(theta)

sp3.grad

$\displaystyle \mathbf{e}_{r} \frac{\partial}{\partial r} + \mathbf{e}_{\phi} \frac{1}{r} \frac{\partial}{\partial \phi} + \mathbf{e}_{\theta} \frac{1}{r \sin{\left(\phi \right)}} \frac{\partial}{\partial \theta}$