notebook.community

Edit and run

This example is kindly contributed by FreddyBaudine for reproducing pygae/galgebra#26 and pygae/galgebra#30 with modifications by utensil.

Please note before Python code, there's an invisible markdown cell with the following code to enable color and define some colors from http://latexcolor.com/:

$$
\require{color}
\definecolor{airforceblue}{rgb}{0.36, 0.54, 0.66}
\definecolor{applegreen}{rgb}{0.55, 0.71, 0.0}
\definecolor{atomictangerine}{rgb}{1.0, 0.6, 0.4}
$$

$$ \require{color} \definecolor{airforceblue}{rgb}{0.36, 0.54, 0.66} \definecolor{applegreen}{rgb}{0.55, 0.71, 0.0} \definecolor{atomictangerine}{rgb}{1.0, 0.6, 0.4} $$



In [1]:

    
from __future__ import print_function
import sys
from galgebra.printer import  Format, xpdf
Format()
from sympy import symbols, sin, pi, latex, Array, permutedims
from galgebra.ga import Ga



In [2]:

    
from IPython.display import Math

Base manifold (three dimensional)

Metric tensor (cartesian coordinates - norm = False)



In [3]:

    
from sympy import cos, sin, symbols
g3coords = (x,y,z) = symbols('x y z')
g3 = Ga('ex ey ez', g = [1,1,1], coords = g3coords,norm=False) # Create g3
(e_x,e_y,e_z) = g3.mv()



In [4]:

    
Math(r'g =%s' % latex(g3.g))









    Out[4]:





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

Two dimensioanal submanifold - Unit sphere

Basis not normalised



In [5]:

    
sp2coords = (theta, phi) = symbols(r'{\color{airforceblue}\theta} {\color{applegreen}\phi}', real = True)
sp2param = [sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)]

sp2 = g3.sm(sp2param, sp2coords, norm = False) # submanifold

(etheta, ephi) = sp2.mv() # sp2 basis vectors
(rtheta, rphi) = sp2.mvr() # sp2 reciprocal basis vectors

sp2grad = sp2.grad

sph_map = [1, theta, phi]  # Coordinate map for sphere of r = 1



In [6]:

    
Math(r'(\theta,\phi)\rightarrow (r,\theta,\phi) = %s' % latex(sph_map))









    Out[6]:





$\displaystyle (\theta,\phi)\rightarrow (r,\theta,\phi) = \left[ 1, \  {\color{airforceblue}\theta}, \  {\color{applegreen}\phi}\right]$



In [7]:

    
Math(r'e_\theta \cdot e_\theta = %s' % (etheta|etheta))









    Out[7]:





$\displaystyle e_\theta \cdot e_\theta = 1$



In [8]:

    
Math(r'e_\phi \cdot e_\phi = %s' % (ephi|ephi))









    Out[8]:





$\displaystyle e_\phi \cdot e_\phi = {\sin{\left ({\color{airforceblue}\theta} \right )}}^{2}$



In [9]:

    
Math('g = %s' % latex(sp2.g))









    Out[9]:





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



In [10]:

    
Math(r'g^{-1} = %s' % latex(sp2.g_inv))









    Out[10]:





$\displaystyle g^{-1} = \left[\begin{matrix}1 & 0\\0 & \frac{1}{\sin^{2}{\left({\color{airforceblue}\theta} \right)}}\end{matrix}\right]$

Christoffel symbols of the first kind:



In [11]:

    
Cf1 = sp2.Christoffel_symbols(mode=1)
Cf1 = permutedims(Array(Cf1), (2, 0, 1))



In [12]:

    
Math(r'\Gamma_{1, \alpha, \beta} = %s \quad \Gamma_{2, \alpha, \beta} = %s ' % (latex(Cf1[0, :, :]), latex(Cf1[1, :, :])))









    Out[12]:





$\displaystyle \Gamma_{1, \alpha, \beta} = \left[\begin{matrix}0 & 0\\0 & - \frac{\sin{\left(2 {\color{airforceblue}\theta} \right)}}{2}\end{matrix}\right] \quad \Gamma_{2, \alpha, \beta} = \left[\begin{matrix}0 & \frac{\sin{\left(2 {\color{airforceblue}\theta} \right)}}{2}\\\frac{\sin{\left(2 {\color{airforceblue}\theta} \right)}}{2} & 0\end{matrix}\right] $



In [13]:

    
Cf2 = sp2.Christoffel_symbols(mode=2)
Cf2 = permutedims(Array(Cf2), (2, 0, 1))



In [14]:

    
Math(r'\Gamma^{1}_{\phantom{1,}\alpha, \beta} = %s \quad \Gamma^{2}_{\phantom{2,}\alpha, \beta} = %s ' % (latex(Cf2[0, :, :]), latex(Cf2[1, :, :])))









    Out[14]:





$\displaystyle \Gamma^{1}_{\phantom{1,}\alpha, \beta} = \left[\begin{matrix}0 & 0\\0 & - \frac{\sin{\left(2 {\color{airforceblue}\theta} \right)}}{2}\end{matrix}\right] \quad \Gamma^{2}_{\phantom{2,}\alpha, \beta} = \left[\begin{matrix}0 & \frac{1}{\tan{\left({\color{airforceblue}\theta} \right)}}\\\frac{1}{\tan{\left({\color{airforceblue}\theta} \right)}} & 0\end{matrix}\right] $



In [15]:

    
F = sp2.mv('F','vector',f=True) #scalar function
f = sp2.mv('f','scalar',f=True) #vector function



In [16]:

    
Math(r'\nabla = %s' % sp2grad)









    Out[16]:





$\displaystyle \nabla = \boldsymbol{e}_{{\color{airforceblue}\theta}} \frac{\partial}{\partial {\color{airforceblue}\theta}} + \boldsymbol{e}_{{\color{applegreen}\phi}} \frac{1}{{\sin{\left ({\color{airforceblue}\theta} \right )}}^{2}} \frac{\partial}{\partial {\color{applegreen}\phi}}$



In [17]:

    
Math(r'\nabla f = %s' % (sp2.grad * f))









    Out[17]:





$\displaystyle \nabla f = \partial_{{\color{airforceblue}\theta}} f  \boldsymbol{e}_{{\color{airforceblue}\theta}} + \frac{\partial_{{\color{applegreen}\phi}} f }{{\sin{\left ({\color{airforceblue}\theta} \right )}}^{2}} \boldsymbol{e}_{{\color{applegreen}\phi}}$



In [18]:

    
Math(r'F = %s' % F)









    Out[18]:





$\displaystyle F = F^{{\color{airforceblue}\theta}}  \boldsymbol{e}_{{\color{airforceblue}\theta}} + F^{{\color{applegreen}\phi}}  \boldsymbol{e}_{{\color{applegreen}\phi}}$



In [19]:

    
Math(r'\nabla F = %s' % (sp2.grad * F))









    Out[19]:





$\displaystyle \nabla F = \left ( \frac{F^{{\color{airforceblue}\theta}} }{\tan{\left ({\color{airforceblue}\theta} \right )}} + \partial_{{\color{airforceblue}\theta}} F^{{\color{airforceblue}\theta}}  + \partial_{{\color{applegreen}\phi}} F^{{\color{applegreen}\phi}} \right )  + \left ( \frac{2 F^{{\color{applegreen}\phi}} }{\tan{\left ({\color{airforceblue}\theta} \right )}} + \partial_{{\color{airforceblue}\theta}} F^{{\color{applegreen}\phi}}  - \frac{\partial_{{\color{applegreen}\phi}} F^{{\color{airforceblue}\theta}} }{{\sin{\left ({\color{airforceblue}\theta} \right )}}^{2}}\right ) \boldsymbol{e}_{{\color{airforceblue}\theta}}\wedge \boldsymbol{e}_{{\color{applegreen}\phi}}$

One dimensioanal submanifold

Basis not normalised



In [20]:

    
cir_th = phi = symbols(r'{\color{atomictangerine}\phi}',real = True)
cir_map = [pi/8, phi]



In [21]:

    
Math(r'(\phi)\rightarrow (\theta,\phi) = %s' % latex(cir_map))









    Out[21]:





$\displaystyle (\phi)\rightarrow (\theta,\phi) = \left[ \frac{\pi}{8}, \  {\color{atomictangerine}\phi}\right]$



In [22]:

    
cir1d = sp2.sm( cir_map , (cir_th,), norm = False) # submanifold

cir1dgrad = cir1d.grad

(ephi) = cir1d.mv()



In [23]:

    
Math(r'e_\phi \cdot e_\phi = %s' % latex(ephi[0] | ephi[0]))









    Out[23]:





$\displaystyle e_\phi \cdot e_\phi = \frac{1}{2} - \frac{\sqrt{2}}{4}$



In [24]:

    
Math('g = %s' % latex(cir1d.g))









    Out[24]:





$\displaystyle g = \left[\begin{matrix}\frac{1}{2} - \frac{\sqrt{2}}{4}\end{matrix}\right]$



In [25]:

    
h = cir1d.mv('h','scalar',f= True)

H = cir1d.mv('H','vector',f= True)



In [26]:

    
Math(r'\nabla = %s' % cir1dgrad)









    Out[26]:





$\displaystyle \nabla = \boldsymbol{e}_{{\color{atomictangerine}\phi}} \left ( 2 \sqrt{2} + 4\right ) \frac{\partial}{\partial {\color{atomictangerine}\phi}}$



In [27]:

    
Math(r'\nabla h = %s' %(cir1d.grad * h).simplify())









    Out[27]:





$\displaystyle \nabla h = \left(2 \sqrt{2} + 4\right) \partial_{{\color{atomictangerine}\phi}} h  \boldsymbol{e}_{{\color{atomictangerine}\phi}}$



In [28]:

    
Math('H = %s' % H)









    Out[28]:





$\displaystyle H = H^{{\color{atomictangerine}\phi}}  \boldsymbol{e}_{{\color{atomictangerine}\phi}}$



In [29]:

    
Math(r'\nabla H = %s' % (cir1d.grad * H).simplify())









    Out[29]:





$\displaystyle \nabla H = \partial_{{\color{atomictangerine}\phi}} H^{{\color{atomictangerine}\phi}} $



In [ ]: