def main():
#Get_Program()
Format()
#EM_Waves_in_Geom_Calculus_Complex()
EM_Waves_in_Geom_Calculus_Real()
xpdf()
return
def main():
#Get_Program()
Format()
EM_Waves_in_Geom_Calculus()
xpdf()
return
def main():
Get_Program()
Format()
basic_multivector_operations_3D()
basic_multivector_operations_2D()
xpdf()
return
def main():
Get_Program()
Format()
Maxwells_Equations_in_Geom_Calculus()
Dirac_Equation_in_Geog_Calculus()
Lorentz_Tranformation_in_Geog_Algebra()
xpdf()
return
def main():
Get_Program()
Format()
"""
The uncommented examples take about 3'20" to run on an i7 in turbo
mode (3.8 GHz). I have never waited long enough to run the commented
out examples.
"""
derivatives_in_spherical_coordinates()
derivatives_in_paraboloidal_coordinates()
derivatives_in_elliptic_cylindrical_coordinates()
derivatives_in_prolate_spheroidal_coordinates()
#derivatives_in_oblate_spheroidal_coordinates()
#derivatives_in_bipolar_coordinates()
#derivatives_in_toroidal_coordinates()
xpdf(paper=(20,11))
return
def main():
Get_Program()
Format()
basic_multivector_operations_3D()
basic_multivector_operations_2D()
basic_multivector_operations_2D_orthogonal()
check_generalized_BAC_CAB_formulas()
rounding_numerical_components()
derivatives_in_rectangular_coordinates()
derivatives_in_spherical_coordinates()
noneuclidian_distance_calculation()
conformal_representations_of_circles_lines_spheres_and_planes()
properties_of_geometric_objects()
extracting_vectors_from_conformal_2_blade()
reciprocal_frame_test()
xpdf()
return
print '#2d general ($A,\\;B$ are linear transformations)'
A2d = g2d.lt('A')
print 'A =', A2d
print '\\f{\\det}{A} =', A2d.det()
#A2d.adj().Fmt(4,'\\overline{A}')
print '\\f{\\Tr}{A} =', A2d.tr()
print '\\f{A}{e_u^e_v} =', A2d(eu^ev)
print '\\f{A}{e_u}^\\f{A}{e_v} =', A2d(eu)^A2d(ev)
B2d = g2d.lt('B')
print 'B =', B2d
print 'A + B =', A2d + B2d
print 'AB =', A2d * B2d
print 'A - B =', A2d - B2d
a = g2d.mv('a','vector')
b = g2d.mv('b','vector')
print r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',((a|A2d.adj()(b))-(b|A2d(a))).simplify()
print '#4d Minkowski spaqce (Space Time)'
m4d = Ga('e_t e_x e_y e_z', g=[1, -1, -1, -1],coords=symbols('t,x,y,z',real=True))
T = m4d.lt('T')
print 'g =', m4d.g
print r'\underline{T} =',T
print r'\overline{T} =',T.adj()
#m4d.mv(T.det()).Fmt(4,r'\f{\det}{\underline{T}}')
print r'\f{\mbox{tr}}{\underline{T}} =',T.tr()
a = m4d.mv('a','vector')
b = m4d.mv('b','vector')
print r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',((a|T.adj()(b))-(b|T(a))).simplify()
xpdf(paper='landscape')
import sys
from sympy import symbols,sin,cos
from sympy.galgebra.printer import Format,xpdf,Get_Program,Print_Function
from sympy.galgebra.ga import Ga
Format()
coords = symbols('t x y z',real=True)
(st4d,g0,g1,g2,g3) = Ga.build('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=coords)
I = st4d.i
(m,e) = symbols('m e')
psi = st4d.mv('psi','spinor',f=True)
A = st4d.mv('A','vector',f=True)
sig_z = g3*g0
print '\\text{4-Vector Potential\\;\\;}\\bm{A} =',A
print '\\text{8-component real spinor\\;\\;}\\bm{\\psi} =',psi
dirac_eq = (st4d.grad*psi)*I*sig_z-e*A*psi-m*psi*g0
dirac_eq = dirac_eq.simplify()
dirac_eq.Fmt(3,r'%\text{Dirac Equation\;\;}\nabla \bm{\psi}'+\
r' I \sigma_{z}-e\bm{A}\bm{\psi}-m\bm{\psi}\gamma_{t} = 0')
xpdf(paper='landscape',prog=True)
w = E2 | e1
w = w.expand()
print "E2|e1 =", w
w = E2 | e3
w = w.expand()
print "E2|e3 =", w
w = E3 | e1
w = w.expand()
print "E3|e1 =", w
w = E3 | e2
w = w.expand()
print "E3|e2 =", w
w = E1 | e1
w = (w.expand()).scalar()
Esq = expand(Esq)
print "%(E1\\cdot e1)/E^{2} =", simplify(w / Esq)
w = E2 | e2
w = (w.expand()).scalar()
print "%(E2\\cdot e2)/E^{2} =", simplify(w / Esq)
w = E3 | e3
w = (w.expand()).scalar()
print "%(E3\\cdot e3)/E^{2} =", simplify(w / Esq)
xpdf(paper="letter")
print '\\f{\\det}{A} =', A2d.det()
print '\\f{\\Tr}{A} =', A2d.tr()
B2d = g2d.lt('B')
print 'B =', B2d
print 'A + B =', A2d + B2d
print 'AB =', A2d * B2d
print 'A - B =', A2d - B2d
a = g2d.mv('a','vector')
b = g2d.mv('b','vector')
print r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',((a|A2d.adj()(b))-(b|A2d(a))).simplify()
m4d = Ga('e_t e_x e_y e_z', g=[1, -1, -1, -1],coords=symbols('t,x,y,z',real=True))
T = m4d.lt('T')
print '#$T$ is a linear transformation in Minkowski space'
print r'\underline{T} =',T
print r'\overline{T} =',T.adj()
print r'\f{\mbox{tr}}{\underline{T}} =',T.tr()
a = m4d.mv('a','vector')
b = m4d.mv('b','vector')
print r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',((a|T.adj()(b))-(b|T(a))).simplify()
xpdf(paper='landscape',crop=True)
print 'grad*F =',grad_uv * F
tp = (th,phi) = symbols('theta,phi',real=True)
smap = [sin(th)*cos(phi),sin(th)*sin(phi),cos(th)]
sph2dr = o3d.sm(smap,tp,norm=True)
(eth, ephi) = sph2dr.mv()
grad_tp = sph2dr.grad
F = sph2dr.mv('F','vector',f=True)
f = sph2dr.mv('f','scalar',f=True)
print 'f =',f
print 'grad*f =',grad_tp * f
print 'F =',F
print 'grad*F =',grad_tp * F
return
def dummy():
return
if __name__ == "__main__":
Format()
Get_Program()
main()
xpdf(paper=(22,11))
def main():
Format()
(g3d,ex,ey,ez) = Ga.build('e*x|y|z')
A = g3d.mv('A','mv')
print r'\bm{A} =',A
A.Fmt(2,r'\bm{A}')
A.Fmt(3,r'\bm{A}')
X = (x,y,z) = symbols('x y z',real=True)
o3d = Ga('e_x e_y e_z',g=[1,1,1],coords=X)
(ex,ey,ez) = o3d.mv()
f = o3d.mv('f','scalar',f=True)
A = o3d.mv('A','vector',f=True)
B = o3d.mv('B','bivector',f=True)
print r'\bm{A} =',A
print r'\bm{B} =',B
print 'grad*f =',o3d.grad*f
print r'grad|\bm{A} =',o3d.grad|A
print r'grad*\bm{A} =',o3d.grad*A
print r'-I*(grad^\bm{A}) =',-o3d.i*(o3d.grad^A)
print r'grad*\bm{B} =',o3d.grad*B
print r'grad^\bm{B} =',o3d.grad^B
print r'grad|\bm{B} =',o3d.grad|B
g4d = Ga('a b c d')
(a,b,c,d) = g4d.mv()
print 'g_{ij} =',g4d.g
print '\\bm{a|(b*c)} =',a|(b*c)
print '\\bm{a|(b^c)} =',a|(b^c)
print '\\bm{a|(b^c^d)} =',a|(b^c^d)
print '\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',(a|(b^c))+(c|(a^b))+(b|(c^a))
print '\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',a*(b^c)-b*(a^c)+c*(a^b)
print '\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)
print '\\bm{(a^b)|(c^d)} =',(a^b)|(c^d)
print '\\bm{((a^b)|c)|d} =',((a^b)|c)|d
print '\\bm{(a^b)\\times (c^d)} =',Com(a^b,c^d)
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
ng3d = Ga('e1 e2 e3',g=g)
(e1,e2,e3) = ng3d.mv()
E = e1^e2^e3
Esq = (E*E).scalar()
print 'E =',E
print '%E^{2} =',Esq
Esq_inv = 1/Esq
E1 = (e2^e3)*E
E2 = (-1)*(e1^e3)*E
E3 = (e1^e2)*E
print 'E1 = (e2^e3)*E =',E1
print 'E2 =-(e1^e3)*E =',E2
print 'E3 = (e1^e2)*E =',E3
print 'E1|e2 =',(E1|e2).expand()
print 'E1|e3 =',(E1|e3).expand()
print 'E2|e1 =',(E2|e1).expand()
print 'E2|e3 =',(E2|e3).expand()
print 'E3|e1 =',(E3|e1).expand()
print 'E3|e2 =',(E3|e2).expand()
w = ((E1|e1).expand()).scalar()
Esq = expand(Esq)
print '%(E1\\cdot e1)/E^{2} =',simplify(w/Esq)
w = ((E2|e2).expand()).scalar()
print '%(E2\\cdot e2)/E^{2} =',simplify(w/Esq)
w = ((E3|e3).expand()).scalar()
print '%(E3\\cdot e3)/E^{2} =',simplify(w/Esq)
X = (r,th,phi) = symbols('r theta phi')
s3d = Ga('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=X,norm=True)
(er,eth,ephi) = s3d.mv()
f = s3d.mv('f','scalar',f=True)
A = s3d.mv('A','vector',f=True)
B = s3d.mv('B','bivector',f=True)
print 'A =',A
print 'B =',B
print 'grad*f =',s3d.grad*f
print 'grad|A =',s3d.grad|A
print '-I*(grad^A) =',-s3d.i*(s3d.grad^A)
print 'grad^B =',s3d.grad^B
coords = symbols('t x y z')
m4d = Ga('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=coords)
(g0,g1,g2,g3) = m4d.mv()
I = m4d.i
B = m4d.mv('B','vector',f=True)
E = m4d.mv('E','vector',f=True)
B.set_coef(1,0,0)
E.set_coef(1,0,0)
B *= g0
E *= g0
J = m4d.mv('J','vector',f=True)
F = E+I*B
print 'B = \\bm{B\\gamma_{t}} =',B
print 'E = \\bm{E\\gamma_{t}} =',E
print 'F = E+IB =',F
print 'J =',J
gradF = m4d.grad*F
gradF.Fmt(3,'grad*F')
print 'grad*F = J'
(gradF.get_grade(1)-J).Fmt(3,'%\\grade{\\nabla F}_{1} -J = 0')
(gradF.get_grade(3)).Fmt(3,'%\\grade{\\nabla F}_{3} = 0')
(alpha,beta,gamma) = symbols('alpha beta gamma')
(x,t,xp,tp) = symbols("x t x' t'")
m2d = Ga('gamma*t|x',g=[1,-1])
(g0,g1) = m2d.mv()
R = cosh(alpha/2)+sinh(alpha/2)*(g0^g1)
X = t*g0+x*g1
Xp = tp*g0+xp*g1
print 'R =',R
print r"#%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} = t'\bm{\gamma'_{t}}+x'\bm{\gamma'_{x}} = R\lp t'\bm{\gamma_{t}}+x'\bm{\gamma_{x}}\rp R^{\dagger}"
Xpp = R*Xp*R.rev()
Xpp = Xpp.collect()
Xpp = Xpp.trigsimp()
print r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =",Xpp
Xpp = Xpp.subs({sinh(alpha):gamma*beta,cosh(alpha):gamma})
print r'%\f{\sinh}{\alpha} = \gamma\beta'
print r'%\f{\cosh}{\alpha} = \gamma'
print r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =",Xpp.collect()
coords = symbols('t x y z')
m4d = Ga('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=coords)
(g0,g1,g2,g3) = m4d.mv()
I = m4d.i
(m,e) = symbols('m e')
psi = m4d.mv('psi','spinor',f=True)
A = m4d.mv('A','vector',f=True)
sig_z = g3*g0
print '\\bm{A} =',A
print '\\bm{\\psi} =',psi
dirac_eq = (m4d.grad*psi)*I*sig_z-e*A*psi-m*psi*g0
dirac_eq.simplify()
dirac_eq.Fmt(3,r'\nabla \bm{\psi} I \sigma_{z}-e\bm{A}\bm{\psi}-m\bm{\psi}\gamma_{t} = 0')
xpdf()
return
print '#2d general ($A,\\;B$ are linear transformations)'
A2d = g2d.lt('A')
print 'A =', A2d
print '\\f{\\det}{A} =', A2d.det()
#A2d.adj().Fmt(4,'\\overline{A}')
print '\\f{\\Tr}{A} =', A2d.tr()
print '\\f{A}{e_u^e_v} =', A2d(eu^ev)
print '\\f{A}{e_u}^\\f{A}{e_v} =', A2d(eu)^A2d(ev)
B2d = g2d.lt('B')
print 'B =', B2d
print 'A + B =', A2d + B2d
print 'AB =', A2d * B2d
print 'A - B =', A2d - B2d
a = g2d.mv('a','vector')
b = g2d.mv('b','vector')
print r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',((a|A2d.adj()(b))-(b|A2d(a))).simplify()
print '#4d Minkowski spaqce (Space Time)'
m4d = Ga('e_t e_x e_y e_z', g=[1, -1, -1, -1],coords=symbols('t,x,y,z',real=True))
T = m4d.lt('T')
print 'g =', m4d.g
print r'\underline{T} =',T
print r'\overline{T} =',T.adj()
#m4d.mv(T.det()).Fmt(4,r'\f{\det}{\underline{T}}')
print r'\f{\mbox{tr}}{\underline{T}} =',T.tr()
a = m4d.mv('a','vector')
b = m4d.mv('b','vector')
print r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',((a|T.adj()(b))-(b|T(a))).simplify()
xpdf(paper=(10,12))
from sympy import symbols, sin
from sympy.galgebra.printer import Format, xpdf
from sympy.galgebra.ga import Ga
Format()
X = (r,th,phi) = symbols('r theta phi')
s3d = Ga('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=X,norm=True)
(er,eth,ephi) = s3d.mv()
grad = s3d.grad
f = s3d.mv('f','scalar',f=True)
A = s3d.mv('A','vector',f=True)
B = s3d.mv('B','bivector',f=True)
print 'f =',f
print 'A =',A
print 'B =',B
print 'grad*f =',grad*f
print 'grad|A =',grad|A
print '-I*(grad^A) =',(-s3d.i*(grad^A)).simplify()
print 'grad^B =',grad^B
xpdf(paper='letter')
self.Fij = []
self.Ki = []
for i in range(n):
self.Ki.append(self.e[i] * self.ebar[i])
print r'%F_{'+str(i)+'} =',self.Ki[-1] * self.Ki[-1].rev()
for j in range(i):
self.Eij.append(self.e[i] * self.e[j] - self.ebar[i] * self.ebar[j])
self.Fij.append(self.e[i] * self.ebar[j] - self.ebar[i] * self.e[j])
print r'%E_{'+str(i)+str(j)+'} =',self.Eij[-1] * self.Eij[-1].rev()
print r'%F_{'+str(i)+str(j)+'} =',self.Fij[-1] * self.Fij[-1].rev()
print 'K_{i} =',self.Ki
print 'E_{ij} =',self.Eij
print 'F_{ij} =',self.Fij
E = self.Eij[0]/2
for i in range(2*n):
print E
E *= self.Eij[0]/2
if __name__ == "__main__":
Format()
la = LieAlgebra(6)
print la.e
print la.ebar
xpdf()
Format()
coords = (r, th, phi) = symbols('r,theta,phi', real=True)
sp3d = Ga('e_r e_th e_ph', g=[1, r**2, r**2*sin(th)**2], \
coords=coords, norm=True)
sph_uv = (u, v) = symbols('u,v', real=True)
sph_map = [1, u, v] # Coordinate map for sphere of r = 1
sph2d = sp3d.sm(sph_map,sph_uv)
print r'(u,v)\rightarrow (r,\theta,\phi) = ',latex(sph_map)
print 'g =',latex(sph2d.g)
F = sph2d.mv('F','vector',f=True) #scalar function
f = sph2d.mv('f','scalar',f=True) #vector function
print r'\nabla f =',sph2d.grad * f
print 'F =',F
print r'\nabla F = ',sph2d.grad * F
cir_s = s = symbols('s',real=True)
cir_map = [pi/8,s]
cir1d = sph2d.sm(cir_map,(cir_s,))
print 'g =',latex(cir1d.g)
h = cir1d.mv('h','scalar',f=True)
H = cir1d.mv('H','vector',f=True)
print r'(s)\rightarrow (u,v) = ',latex(cir_map)
print 'H =', H
print latex(H)
print r'\nabla h =', cir1d.grad * h
print r'\nabla H =', cir1d.grad * H
xpdf(filename='submanifold.tex',paper=(6,5),crop=True)
import sys
from sympy import symbols,sin
from sympy.galgebra.printer import Format,xpdf
from sympy.galgebra.ga import Ga
Format()
X = (r,th,phi) = symbols('r theta phi')
s3d = Ga('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=X,norm=True)
(er,eth,ephi) = s3d.mv()
grad = s3d.grad
f = s3d.mv('f','scalar',f=True)
A = s3d.mv('A','vector',f=True)
B = s3d.mv('B','bivector',f=True)
print 'f =',f
print 'A =',A
print 'B =',B
print 'grad*f =',grad*f
print 'grad|A =',grad|A
print '-I*(grad^A) =',(-s3d.i*(grad^A)).simplify()
print 'grad^B =',grad^B
print r'%\nabla\cdot\lp\nabla f\rp =', s3d.grad|(s3d.grad*f)
xpdf(paper='letter',prog=True)
print r'grad =', o3d.grad
print r'%\nabla^{2} = \nabla\cdot\nabla =', lap
print r'%\lp\nabla^{2}\rp f =', lap*f
print r'%\nabla\cdot\lp\nabla f\rp =', o3d.grad | (o3d.grad * f)
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)
f = sp3d.mv('f', 'scalar', f=True)
lap = sp3d.grad*sp3d.grad
print r'%\nabla =', sp3d.grad
print r'%\nabla^{2} = \nabla\cdot\nabla =', lap
print r'%\lp\nabla^{2}\rp f =', lap*f
print r'%\nabla\cdot\lp\nabla f\rp =', sp3d.grad | (sp3d.grad * f)
A = o3d.mv('A','vector')
K = o3d.mv('K','vector')
xs = o3d.mv(x)
X = o3d.X()
print o3d.grad*A
print A*o3d.grad
print o3d.grad*xs
print xs*o3d.grad
print o3d.grad*(o3d.grad+xs)
print (o3d.grad+xs)*o3d.grad
print X
print X^o3d.grad
print (X^o3d.grad).components()
xpdf(filename='prod.tex')
Format()
coords = (x, y, z) = symbols("x y z", real=True)
(o3d, ex, ey, ez) = Ga.build("e*x|y|z", g=[1, 1, 1], coords=coords)
X = x * ex + y * ey + z * ez
I = o3d.i
v = o3d.mv("v", "vector")
f = o3d.mv("f", "scalar", f=True)
A = o3d.mv("A", "vector", f=True)
dd = v | o3d.grad
lap = o3d.grad * o3d.grad
print r"\bm{X} =", X
print r"\bm{v} =", v
print r"\bm{A} =", A
print r"%\bm{v}\cdot\nabla =", dd
print r"%\nabla^{2} =", lap
print r"%\bm{v}\cdot\nabla f =", dd * f
print r"%\nabla^{2} f =", lap * f
print r"%\nabla^{2} \bm{A} =", lap * A
print r"%\bar{\nabla}\cdot v =", o3d.rgrad | v
Xgrad = X | o3d.grad
rgradX = o3d.rgrad | X
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)
f = sp3d.mv("f", "scalar", f=True)
lap = sp3d.grad * sp3d.grad
print r"%\nabla^{2} = \nabla\cdot\nabla =", lap
print r"%\lp\nabla^{2}\rp f =", lap * f
print r"%\nabla\cdot\lp\nabla f\rp =", sp3d.grad | (sp3d.grad * f)
xpdf(paper="landscape", crop=True)
return
Format()
(glg,ei,ej,em,en,ebi,ebj,ebm,ebn) = Ga.build(r'e_i e_j e_k e_l \bar{e}_i \bar{e}_j \bar{e}_k \bar{e}_l',g=[1,1,1,1,-1,-1,-1,-1])
e = {'i':ei,'j':ej,'k':em,'l':en}
eb = {'i':ebi,'j':ebj,'k':ebm,'l':ebn}
print r'#\center{General Linear Group of Order $n$\newline}'
print r'#Lie Algebra Generators: $1\le i < j \le n$ and $1 \le i < l \le n$'
Eij = E('i','j')
Fij = F('i','j')
Ki = K('i')
Eil = E('i','l')
Fil = F('i','l')
print r'#Non Zero Commutators'
ComP(Eij,Fij)
ComP(Eij,Ki)
ComP(Fij,Ki)
ComP(Eij,Eil)
ComP(Fij,Fil)
ComP(Fij,Eil)
xpdf(paper='letter',pt='12pt',debug=True,prog=True)