def Fmt_test():
Print_Function()
e3d = Ga('e1 e2 e3',g=[1,1,1])
v = e3d.mv('v','vector')
B = e3d.mv('B','bivector')
M = e3d.mv('M','mv')
Fmt(2)
print '#Global $Fmt = 2$'
print 'v =',v
print 'B =',B
print 'M =',M
print '#Using $.Fmt()$ Function'
print 'v.Fmt(3) =',v.Fmt(3)
print 'B.Fmt(3) =',B.Fmt(3)
print 'M.Fmt(2) =',M.Fmt(2)
print 'M.Fmt(1) =',M.Fmt(1)
print '#Global $Fmt = 1$'
Fmt(1)
print 'v =',v
print 'B =',B
print 'M =',M
return
def extracting_vectors_from_conformal_2_blade():
Print_Function()
print r'B = P1\W P2'
g = '0 -1 #,'+ \
'-1 0 #,'+ \
'# # #'
c2b = Ga('P1 P2 a',g=g)
(P1,P2,a) = c2b.mv()
print 'g_{ij} =',c2b.g
B = P1^P2
Bsq = B*B
print '%B^{2} =',Bsq
ap = a-(a^B)*B
print "a' = a-(a^B)*B =",ap
Ap = ap+ap*B
Am = ap-ap*B
print "A+ = a'+a'*B =",Ap
print "A- = a'-a'*B =",Am
print '%(A+)^{2} =',Ap*Ap
print '%(A-)^{2} =',Am*Am
aB = a|B
print 'a|B =',aB
return
def derivatives_in_rectangular_coordinates():
Print_Function()
X = (x,y,z) = symbols('x y z')
o3d = Ga('e_x e_y e_z',g=[1,1,1],coords=X)
(ex,ey,ez) = o3d.mv()
grad = o3d.grad
f = o3d.mv('f','scalar',f=True)
A = o3d.mv('A','vector',f=True)
B = o3d.mv('B','bivector',f=True)
C = o3d.mv('C','mv')
print 'f =',f
print 'A =',A
print 'B =',B
print 'C =',C
print 'grad*f =',grad*f
print 'grad|A =',grad|A
print 'grad*A =',grad*A
print '-I*(grad^A) =',-o3d.i*(grad^A)
print 'grad*B =',grad*B
print 'grad^B =',grad^B
print 'grad|B =',grad|B
return
def basic_multivector_operations_3D():
Print_Function()
g3d = Ga('e*x|y|z')
(ex,ey,ez) = g3d.mv()
A = g3d.mv('A','mv')
A.Fmt(1,'A')
A.Fmt(2,'A')
A.Fmt(3,'A')
A.even().Fmt(1,'%A_{+}')
A.odd().Fmt(1,'%A_{-}')
X = g3d.mv('X','vector')
Y = g3d.mv('Y','vector')
print 'g_{ij} = ',g3d.g
X.Fmt(1,'X')
Y.Fmt(1,'Y')
(X*Y).Fmt(2,'X*Y')
(X^Y).Fmt(2,'X^Y')
(X|Y).Fmt(2,'X|Y')
return
def properties_of_geometric_objects():
Print_Function()
global n, nbar
g = '# # # 0 0,'+ \
'# # # 0 0,'+ \
'# # # 0 0,'+ \
'0 0 0 0 2,'+ \
'0 0 0 2 0'
c3d = Ga('p1 p2 p3 n \\bar{n}',g=g)
(p1,p2,p3,n,nbar) = c3d.mv()
print 'g_{ij} =',c3d.g
P1 = F(p1)
P2 = F(p2)
P3 = F(p3)
print '\\text{Extracting direction of line from }L = P1\\W P2\\W n'
L = P1^P2^n
delta = (L|n)|nbar
print '(L|n)|\\bar{n} =',delta
print '\\text{Extracting plane of circle from }C = P1\\W P2\\W P3'
C = P1^P2^P3
delta = ((C^n)|n)|nbar
print '((C^n)|n)|\\bar{n}=',delta
print '(p2-p1)^(p3-p1)=',(p2-p1)^(p3-p1)
return
def test_3_4(self):
"""
The other contraction.
"""
Ga.dual_mode("Iinv+")
R = Ga('e*1|2|3')
A_blades = [R.mv('A', i, 'grade') for i in range(R.n + 1)]
B_blades = [R.mv('B', i, 'grade') for i in range(R.n + 1)]
for A, B in product(A_blades, B_blades):
self.assertEquals(B > A, ((-1) ** (A.pure_grade() * (B.pure_grade() - 1))) * (A < B))
def rounding_numerical_components():
Print_Function()
o3d = Ga('e_x e_y e_z',g=[1,1,1])
(ex,ey,ez) = o3d.mv()
X = 1.2*ex+2.34*ey+0.555*ez
Y = 0.333*ex+4*ey+5.3*ez
print 'X =',X
print 'Nga(X,2) =',Nga(X,2)
print 'X*Y =',X*Y
print 'Nga(X*Y,2) =',Nga(X*Y,2)
return
def main():
Print_Function()
(a, b, c) = abc = symbols("a,b,c", real=True)
(o3d, ea, eb, ec) = Ga.build("e_a e_b e_c", g=[1, 1, 1], coords=abc)
grad = o3d.grad
x = symbols("x", real=True)
A = o3d.lt(
[
[x * a * c ** 2, x ** 2 * a * b * c, x ** 2 * a ** 3 * b ** 5],
[x ** 3 * a ** 2 * b * c, x ** 4 * a * b ** 2 * c ** 5, 5 * x ** 4 * a * b ** 2 * c],
[x ** 4 * a * b ** 2 * c ** 4, 4 * x ** 4 * a * b ** 2 * c ** 2, 4 * x ** 4 * a ** 5 * b ** 2 * c],
]
)
print "A =", A
v = a * ea + b * eb + c * ec
print "v =", v
f = v | A(v)
print r"%f = v\cdot \f{A}{v} =", f
(grad * f).Fmt(3, r"%\nabla f")
Av = A(v)
print r"%\f{A}{v} =", Av
(grad * Av).Fmt(3, r"%\nabla \f{A}{v}")
return
def derivatives_in_prolate_spheroidal_coordinates():
#Print_Function()
a = symbols('a', real=True)
coords = (xi, eta, phi) = symbols('xi eta phi', real=True)
(ps3d, er, eth, ephi) = Ga.build('e_xi e_eta e_phi',
X=[
a * sinh(xi) * sin(eta) * cos(phi),
a * sinh(xi) * sin(eta) * sin(phi),
a * cosh(xi) * cos(eta)
],
coords=coords,
norm=True)
grad = ps3d.grad
f = ps3d.mv('f', 'scalar', f=True)
A = ps3d.mv('A', 'vector', f=True)
B = ps3d.mv('B', 'bivector', f=True)
print('f =', f)
print('A =', A)
print('B =', B)
print('grad*f =', grad * f)
print('grad|A =', grad | A)
(-ps3d.i * (grad ^ A)).Fmt(3, '-I*(grad^A)')
(grad ^ B).Fmt(3, 'grad^B')
return
def derivatives_in_spherical_coordinates():
#Print_Function()
coords = (r, th, phi) = symbols('r theta phi', real=True)
(sp3d, er, eth, ephi) = Ga.build('e_r e_theta e_phi',
g=[1, r**2, r**2 * sin(th)**2],
coords=coords)
grad = sp3d.grad
f = sp3d.mv('f', 'scalar', f=True)
A = sp3d.mv('A', 'vector', f=True)
B = sp3d.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('grad\\times A = -I*(grad^A) =', -sp3d.i * (grad ^ A))
print('%\\nabla^{2}f =', grad | (grad * f))
print('grad^B =', grad ^ B)
"""
print '( \\nabla\\W\\nabla )\\bm{e}_{r} =',((grad^grad)*er).trigsimp()
print '( \\nabla\\W\\nabla )\\bm{e}_{\\theta} =',((grad^grad)*eth).trigsimp()
print '( \\nabla\\W\\nabla )\\bm{e}_{\\phi} =',((grad^grad)*ephi).trigsimp()
"""
return
def derivatives_in_paraboloidal_coordinates():
#Print_Function()
coords = (u, v, phi) = symbols('u v phi', real=True)
(par3d, er, eth, ephi) = Ga.build(
'e_u e_v e_phi',
X=[u * v * cos(phi), u * v * sin(phi), (u**2 - v**2) / 2],
coords=coords,
norm=True)
grad = par3d.grad
f = par3d.mv('f', 'scalar', f=True)
A = par3d.mv('A', 'vector', f=True)
B = par3d.mv('B', 'bivector', f=True)
print('#Derivatives in Paraboloidal Coordinates')
print('f =', f)
print('A =', A)
print('B =', B)
print('grad*f =', grad * f)
print('grad|A =', grad | A)
(-par3d.i * (grad ^ A)).Fmt(3, 'grad\\times A = -I*(grad^A)')
print('grad^B =', grad ^ B)
return
def Lorentz_Tranformation_in_Geog_Algebra():
Print_Function()
(alpha,beta,gamma) = symbols('alpha beta gamma')
(x,t,xp,tp) = symbols("x t x' t'",real=True)
(st2d,g0,g1) = Ga.build('gamma*t|x',g=[1,-1])
from sympy import sinh,cosh
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())
return
def basic_multivector_operations_3D():
#Print_Function()
(g3d, ex, ey, ez) = Ga.build('e*x|y|z')
print 'g_{ij} =', g3d.g
A = g3d.mv('A', 'mv')
A.Fmt(1, 'A')
A.Fmt(2, 'A')
A.Fmt(3, 'A')
A.even().Fmt(1, '%A_{+}')
A.odd().Fmt(1, '%A_{-}')
X = g3d.mv('X', 'vector')
Y = g3d.mv('Y', 'vector')
X.Fmt(1, 'X')
Y.Fmt(1, 'Y')
(X * Y).Fmt(2, 'X*Y')
(X ^ Y).Fmt(2, 'X^Y')
(X | Y).Fmt(2, 'X|Y')
return
def Maxwells_Equations_in_Geom_Calculus():
Print_Function()
X = 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=X)
I = st4d.i
B = st4d.mv("B", "vector", f=True)
E = st4d.mv("E", "vector", f=True)
B.set_coef(1, 0, 0)
E.set_coef(1, 0, 0)
B *= g0
E *= g0
J = st4d.mv("J", "vector", f=True)
F = E + I * B
print r"\text{Pseudo Scalar\;\;}I =", I
print "\\text{Magnetic Field Bi-Vector\\;\\;} B = \\bm{B\\gamma_{t}} =", B
print "\\text{Electric Field Bi-Vector\\;\\;} E = \\bm{E\\gamma_{t}} =", E
print "\\text{Electromagnetic Field Bi-Vector\\;\\;} F = E+IB =", F
print "%\\text{Four Current Density\\;\\;} J =", J
gradF = st4d.grad * F
print "#Geom Derivative of Electomagnetic Field Bi-Vector"
gradF.Fmt(3, "grad*F")
print "#Maxwell Equations"
print "grad*F = J"
print "#Div $E$ and Curl $H$ Equations"
print (gradF.get_grade(1) - J).Fmt(3, "%\\grade{\\nabla F}_{1} -J = 0")
print "#Curl $E$ and Div $B$ equations"
print (gradF.get_grade(3)).Fmt(3, "%\\grade{\\nabla F}_{3} = 0")
return
def derivatives_in_spherical_coordinates():
#Print_Function()
coords = (r,th,phi) = symbols('r theta phi', real=True)
(sp3d,er,eth,ephi) = Ga.build('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=coords)
grad = sp3d.grad
f = sp3d.mv('f','scalar',f=True)
A = sp3d.mv('A','vector',f=True)
B = sp3d.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 'grad\\times A = -I*(grad^A) =',-sp3d.i*(grad^A)
print '%\\nabla^{2}f =',grad|(grad*f)
print 'grad^B =',grad^B
"""
print '( \\nabla\\W\\nabla )\\bm{e}_{r} =',((grad^grad)*er).trigsimp()
print '( \\nabla\\W\\nabla )\\bm{e}_{\\theta} =',((grad^grad)*eth).trigsimp()
print '( \\nabla\\W\\nabla )\\bm{e}_{\\phi} =',((grad^grad)*ephi).trigsimp()
"""
return
def test_blade_coefs(self):
"""
Various tests on several multivectors.
"""
(_g3d, e_1, e_2, e_3) = Ga.build('e*1|2|3')
m0 = 2 * e_1 + e_2 - e_3 + 3 * (e_1 ^ e_3) + (e_1 ^ e_3) + (e_2 ^
(3 * e_3))
self.assertTrue(m0.blade_coefs([e_1]) == [2])
self.assertTrue(m0.blade_coefs([e_2]) == [1])
self.assertTrue(m0.blade_coefs([e_1, e_2]) == [2, 1])
self.assertTrue(m0.blade_coefs([e_1 ^ e_3]) == [4])
self.assertTrue(m0.blade_coefs([e_1 ^ e_3, e_2 ^ e_3]) == [4, 3])
self.assertTrue(m0.blade_coefs([e_2 ^ e_3, e_1 ^ e_3]) == [3, 4])
self.assertTrue(m0.blade_coefs([e_1, e_2 ^ e_3]) == [2, 3])
a = Symbol('a')
b = Symbol('b')
m1 = a * e_1 + e_2 - e_3 + b * (e_1 ^ e_2)
self.assertTrue(m1.blade_coefs([e_1]) == [a])
self.assertTrue(m1.blade_coefs([e_2]) == [1])
self.assertTrue(m1.blade_coefs([e_3]) == [-1])
self.assertTrue(m1.blade_coefs([e_1 ^ e_2]) == [b])
self.assertTrue(m1.blade_coefs([e_2 ^ e_3]) == [0])
self.assertTrue(m1.blade_coefs([e_1 ^ e_3]) == [0])
self.assertTrue(m1.blade_coefs([e_1 ^ e_2 ^ e_3]) == [0])
# Invalid parameters
self.assertRaises(ValueError, lambda: m1.blade_coefs([e_1 + e_2]))
self.assertRaises(ValueError, lambda: m1.blade_coefs([e_2 ^ e_1]))
self.assertRaises(ValueError, lambda: m1.blade_coefs([e_1, e_2 ^ e_1]))
self.assertRaises(ValueError, lambda: m1.blade_coefs([a * e_1]))
self.assertRaises(ValueError, lambda: m1.blade_coefs([3 * e_3]))
def main():
Print_Function()
(a, b, c) = abc = symbols('a,b,c', real=True)
(o3d, ea, eb, ec) = Ga.build('e_a e_b e_c', g=[1, 1, 1], coords=abc)
grad = o3d.grad
x = symbols('x', real=True)
A = o3d.lt([[x*a*c**2,x**2*a*b*c,x**2*a**3*b**5],\
[x**3*a**2*b*c,x**4*a*b**2*c**5,5*x**4*a*b**2*c],\
[x**4*a*b**2*c**4,4*x**4*a*b**2*c**2,4*x**4*a**5*b**2*c]])
print('A =', A)
v = a * ea + b * eb + c * ec
print('v =', v)
f = v | A(v)
print(r'%f = v\cdot \f{A}{v} =', f)
(grad * f).Fmt(3, r'%\nabla f')
Av = A(v)
print(r'%\f{A}{v} =', Av)
(grad * Av).Fmt(3, r'%\nabla \f{A}{v}')
return
def Lorentz_Tranformation_in_Geog_Algebra():
Print_Function()
(alpha,beta,gamma) = symbols('alpha beta gamma')
(x,t,xp,tp) = symbols("x t x' t'",real=True)
(st2d,g0,g1) = Ga.build('gamma*t|x',g=[1,-1])
from sympy import sinh,cosh
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()
return
def main():
Eprint()
X = (x,y,z) = symbols('x y z',real=True)
(o3d,ex,ey,ez) = Ga.build('e_x e_y e_z',g=[1,1,1],coords=(x,y,z))
A = x*(ey^ez) + y*(ez^ex) + z*(ex^ey)
print 'A =', A
print 'grad^A =',(o3d.grad^A).simplify()
print
f = o3d.mv(1/sqrt(x**2 + y**2 + z**2))
print 'f =', f
print 'grad*f =',(o3d.grad*f).simplify()
print
B = f*A
print 'B =', B
print
Curl_B = o3d.grad^B
print 'grad^B =', Curl_B.simplify()
return
def test_blade_coefs(self):
"""
Various tests on several multivectors.
"""
(_g3d, e_1, e_2, e_3) = Ga.build('e*1|2|3')
m0 = 2 * e_1 + e_2 - e_3 + 3 * (e_1 ^ e_3) + (e_1 ^ e_3) + (e_2 ^ (3 * e_3))
self.assertTrue(m0.blade_coefs([e_1]) == [2])
self.assertTrue(m0.blade_coefs([e_2]) == [1])
self.assertTrue(m0.blade_coefs([e_1, e_2]) == [2, 1])
self.assertTrue(m0.blade_coefs([e_1 ^ e_3]) == [4])
self.assertTrue(m0.blade_coefs([e_1 ^ e_3, e_2 ^ e_3]) == [4, 3])
self.assertTrue(m0.blade_coefs([e_2 ^ e_3, e_1 ^ e_3]) == [3, 4])
self.assertTrue(m0.blade_coefs([e_1, e_2 ^ e_3]) == [2, 3])
a = Symbol('a')
b = Symbol('b')
m1 = a * e_1 + e_2 - e_3 + b * (e_1 ^ e_2)
self.assertTrue(m1.blade_coefs([e_1]) == [a])
self.assertTrue(m1.blade_coefs([e_2]) == [1])
self.assertTrue(m1.blade_coefs([e_3]) == [-1])
self.assertTrue(m1.blade_coefs([e_1 ^ e_2]) == [b])
self.assertTrue(m1.blade_coefs([e_2 ^ e_3]) == [0])
self.assertTrue(m1.blade_coefs([e_1 ^ e_3]) == [0])
self.assertTrue(m1.blade_coefs([e_1 ^ e_2 ^ e_3]) == [0])
# Invalid parameters
self.assertRaises(ValueError, lambda: m1.blade_coefs([e_1 + e_2]))
self.assertRaises(ValueError, lambda: m1.blade_coefs([e_2 ^ e_1]))
self.assertRaises(ValueError, lambda: m1.blade_coefs([e_1, e_2 ^ e_1]))
self.assertRaises(ValueError, lambda: m1.blade_coefs([a * e_1]))
self.assertRaises(ValueError, lambda: m1.blade_coefs([3 * e_3]))
def Maxwells_Equations_in_Geom_Calculus():
Print_Function()
X = 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=X)
I = st4d.i
B = st4d.mv('B','vector',f=True)
E = st4d.mv('E','vector',f=True)
B.set_coef(1,0,0)
E.set_coef(1,0,0)
B *= g0
E *= g0
J = st4d.mv('J','vector',f=True)
F = E+I*B
print r'\text{Pseudo Scalar\;\;}I =',I
print '\\text{Magnetic Field Bi-Vector\\;\\;} B = \\bm{B\\gamma_{t}} =',B
print '\\text{Electric Field Bi-Vector\\;\\;} E = \\bm{E\\gamma_{t}} =',E
print '\\text{Electromagnetic Field Bi-Vector\\;\\;} F = E+IB =',F
print '%\\text{Four Current Density\\;\\;} J =',J
gradF = st4d.grad*F
print '#Geom Derivative of Electomagnetic Field Bi-Vector'
gradF.Fmt(3,'grad*F')
print '#Maxwell Equations'
print 'grad*F = J'
print '#Div $E$ and Curl $H$ Equations'
(gradF.get_grade(1)-J).Fmt(3,'%\\grade{\\nabla F}_{1} -J = 0')
print '#Curl $E$ and Div $B$ equations'
(gradF.get_grade(3)).Fmt(3,'%\\grade{\\nabla F}_{3} = 0')
return
def test_3_5_4(self):
"""
The duality relationships.
"""
Ga.dual_mode("Iinv+")
R = Ga('e*1|2|3')
A_blades = [R.mv('A', i, 'grade') for i in range(R.n + 1)]
B_blades = [R.mv('B', i, 'grade') for i in range(R.n + 1)]
for A, B in product(A_blades, B_blades):
self.assertEquals((A ^ B).dual(), A < B.dual())
for A, B in product(A_blades, B_blades):
self.assertEquals((A < B).dual(), A ^ B.dual())
def main():
Eprint()
X = (x, y, z) = symbols('x y z', real=True)
(o3d, ex, ey, ez) = Ga.build('e_x e_y e_z', g=[1, 1, 1], coords=(x, y, z))
A = x * (ey ^ ez) + y * (ez ^ ex) + z * (ex ^ ey)
print 'A =', A
print 'grad^A =', (o3d.grad ^ A).simplify()
print
f = o3d.mv(1 / sqrt(x**2 + y**2 + z**2))
print 'f =', f
print 'grad*f =', (o3d.grad * f).simplify()
print
B = f * A
print 'B =', B
print
Curl_B = o3d.grad ^ B
print 'grad^B =', Curl_B.simplify()
return
def Maxwells_Equations_in_Geom_Calculus():
Print_Function()
X = 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=X)
I = st4d.i
B = st4d.mv('B', 'vector', f=True)
E = st4d.mv('E', 'vector', f=True)
B.set_coef(1, 0, 0)
E.set_coef(1, 0, 0)
B *= g0
E *= g0
J = st4d.mv('J', 'vector', f=True)
F = E + I * B
print r'\text{Pseudo Scalar\;\;}I =', I
print '\\text{Magnetic Field Bi-Vector\\;\\;} B = \\bm{B\\gamma_{t}} =', B
print '\\text{Electric Field Bi-Vector\\;\\;} E = \\bm{E\\gamma_{t}} =', E
print '\\text{Electromagnetic Field Bi-Vector\\;\\;} F = E+IB =', F
print '%\\text{Four Current Density\\;\\;} J =', J
gradF = st4d.grad * F
print '#Geom Derivative of Electomagnetic Field Bi-Vector'
gradF.Fmt(3, 'grad*F')
print '#Maxwell Equations'
print 'grad*F = J'
print '#Div $E$ and Curl $H$ Equations'
print(gradF.get_grade(1) - J).Fmt(3, '%\\grade{\\nabla F}_{1} -J = 0')
print '#Curl $E$ and Div $B$ equations'
print(gradF.get_grade(3)).Fmt(3, '%\\grade{\\nabla F}_{3} = 0')
return
def test2_12_2_1(self):
"""
In R2 with Euclidean metric, choose an orthonormal basis {e_1, e_2} in the plane of a and b such that e1 is parallel to a.
Write x = a * e_1 and y = b * (cos(t) * e_1 + sin(t) * e_2), whete t is the angle from a to b.
Evaluate the outer product. What is the geometrical interpretation ?
"""
(_g2d, e_1, e_2) = Ga.build('e*1|2', g='1 0, 0 1')
# TODO: use alpha, beta and theta instead of a, b and t (it crashes sympy)
a = Symbol('a')
b = Symbol('b')
t = Symbol('t')
x = a * e_1
y = b * (cos(t) * e_1 + sin(t) * e_2)
B = x ^ y
self.assertTrue(B == (a * b * sin(t) * (e_1 ^ e_2)))
# Retrieve the parallelogram area from the 2-vector
area = B.norm()
self.assertTrue(area == (a * b * sin(t)))
# Compute the parallelogram area using the determinant
x = [a, 0]
y = [b * cos(t), b * sin(t)]
area = Matrix([x, y]).det()
self.assertTrue(area == (a * b * sin(t)))
def Dirac_Equation_in_Geom_Calculus():
Print_Function()
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()
print dirac_eq.Fmt(
3,
r'%\text{Dirac Equation\;\;}\nabla \bm{\psi} I \sigma_{z}-e\bm{A}\bm{\psi}-m\bm{\psi}\gamma_{t} = 0'
)
return
def basic_multivector_operations_2D():
Print_Function()
g2d = Ga('e*x|y')
(ex,ey) = g2d.mv()
print 'g_{ij} =',g2d.g
X = g2d.mv('X','vector')
A = g2d.mv('A','spinor')
print X.Fmt(1,'X')
print A.Fmt(1,'A')
print (X|A).Fmt(2,'X|A')
print (X<A).Fmt(2,'X<A')
print (A>X).Fmt(2,'A>X')
return
def check_generalized_BAC_CAB_formulas():
Print_Function()
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)
return
def basic_multivector_operations_2D():
Print_Function()
g2d = Ga('e*x|y')
(ex,ey) = g2d.mv()
print 'g_{ij} =',g2d.g
X = g2d.mv('X','vector')
A = g2d.mv('A','spinor')
X.Fmt(1,'X')
A.Fmt(1,'A')
(X|A).Fmt(2,'X|A')
(X<A).Fmt(2,'X<A')
(A>X).Fmt(2,'A>X')
return
def check_generalized_BAC_CAB_formulas():
Print_Function()
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)
return
def derivatives_in_rectangular_coordinates():
Print_Function()
X = (x, y, z) = symbols('x y z')
o3d = Ga('e_x e_y e_z', g=[1, 1, 1], coords=X)
(ex, ey, ez) = o3d.mv()
grad = o3d.grad
f = o3d.mv('f', 'scalar', f=True)
A = o3d.mv('A', 'vector', f=True)
B = o3d.mv('B', 'bivector', f=True)
C = o3d.mv('C', 'mv')
print('f =', f)
print('A =', A)
print('B =', B)
print('C =', C)
print('grad*f =', grad * f)
print('grad|A =', grad | A)
print('grad*A =', grad * A)
print('-I*(grad^A) =', -o3d.I() * (grad ^ A))
print('grad*B =', grad * B)
print('grad^B =', grad ^ B)
print('grad|B =', grad | B)
return
def derivatives_in_rectangular_coordinates():
Print_Function()
X = (x, y, z) = symbols('x y z')
o3d = Ga('e_x e_y e_z', g=[1, 1, 1], coords=X)
(ex, ey, ez) = o3d.mv()
grad = o3d.grad
f = o3d.mv('f', 'scalar', f=True)
A = o3d.mv('A', 'vector', f=True)
B = o3d.mv('B', 'bivector', f=True)
C = o3d.mv('C', 'mv', f=True)
print 'f =', f
print 'A =', A
print 'B =', B
print 'C =', C
print 'grad*f =', grad * f
print 'grad|A =', grad | A
print 'grad*A =', grad * A
print '-I*(grad^A) =', -o3d.E() * (grad ^ A)
print 'grad*B =', grad * B
print 'grad^B =', grad ^ B
print 'grad|B =', grad | B
print 'grad<A =', grad < A
print 'grad>A =', grad > A
print 'grad<B =', grad < B
print 'grad>B =', grad > B
print 'grad<C =', grad < C
print 'grad>C =', grad > C
return
def test_3_5_2(self):
"""
The inverse of a blade.
"""
Ga.dual_mode("Iinv+")
R = Ga('e*1|2|3')
A_blades = [R.mv('A', i, 'grade') for i in range(R.n + 1)]
for A in A_blades:
self.assertEquals(A.inv(), ((-1) ** (A.pure_grade() * (A.pure_grade() - 1) / 2)) * (A / A.norm2()))
for A in A_blades:
self.assertEquals(A < A.inv(), 1)
A = A_blades[1]
self.assertEquals(A.inv(), A / A.norm2())
def General_Lorentz_Tranformation():
Print_Function()
(alpha, beta, gamma) = symbols('alpha beta gamma')
(x, y, z, t) = symbols("x y z t", real=True)
(st4d, g0, g1, g2, g3) = Ga.build('gamma*t|x|y|z', g=[1, -1, -1, -1])
B = (x * g1 + y * g2 + z * g3) ^ (t * g0)
print B
print B.exp(hint='+')
print B.exp(hint='-')
def derivatives_in_spherical_coordinates():
Print_Function()
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
def test2_12_1_3(self):
"""
What is the area of the parallelogram spanned by the vectors a = e_1 + 2*e_2
and b = -e_1 - e_2 (relative to the area of e_1 ^ e_2) ?
"""
(_g3d, e_1, e_2, _e_3) = Ga.build('e*1|2|3')
a = e_1 + 2*e_2
b = -e_1 - e_2
B = a ^ b
self.assertTrue(B == 1 * (e_1 ^ e_2))
def General_Lorentz_Tranformation():
Print_Function()
(alpha, beta, gamma) = symbols("alpha beta gamma")
(x, y, z, t) = symbols("x y z t", real=True)
(st4d, g0, g1, g2, g3) = Ga.build("gamma*t|x|y|z", g=[1, -1, -1, -1])
B = (x * g1 + y * g2 + z * g3) ^ (t * g0)
print B
print B.exp(hint="+")
print B.exp(hint="-")
def conformal_representations_of_circles_lines_spheres_and_planes():
global n, nbar
Print_Function()
g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'
cnfml3d = Ga('e_1 e_2 e_3 n nbar', g=g)
(e1, e2, e3, n, nbar) = cnfml3d.mv()
print 'g_{ij} =\n', cnfml3d.g
e = n + nbar
#conformal representation of points
A = make_vector(e1, ga=cnfml3d) # point a = (1,0,0) A = F(a)
B = make_vector(e2, ga=cnfml3d) # point b = (0,1,0) B = F(b)
C = make_vector(-e1, ga=cnfml3d) # point c = (-1,0,0) C = F(c)
D = make_vector(e3, ga=cnfml3d) # point d = (0,0,1) D = F(d)
X = make_vector('x', 3, ga=cnfml3d)
print 'F(a) =', A
print 'F(b) =', B
print 'F(c) =', C
print 'F(d) =', D
print 'F(x) =', X
print 'a = e1, b = e2, c = -e1, and d = e3'
print 'A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.'
print 'Circle through a, b, and c'
print 'Circle: A^B^C^X = 0 =', (A ^ B ^ C ^ X)
print 'Line through a and b'
print 'Line : A^B^n^X = 0 =', (A ^ B ^ n ^ X)
print 'Sphere through a, b, c, and d'
print 'Sphere: A^B^C^D^X = 0 =', (((A ^ B) ^ C) ^ D) ^ X
print 'Plane through a, b, and d'
print 'Plane : A^B^n^D^X = 0 =', (A ^ B ^ n ^ D ^ X)
L = (A ^ B ^ e) ^ X
print L.Fmt(3, 'Hyperbolic Circle: (A^B^e)^X = 0 =')
return
def conformal_representations_of_circles_lines_spheres_and_planes():
global n,nbar
Print_Function()
g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'
cnfml3d = Ga('e_1 e_2 e_3 n nbar',g=g)
(e1,e2,e3,n,nbar) = cnfml3d.mv()
print 'g_{ij} =\n',cnfml3d.g
e = n+nbar
#conformal representation of points
A = make_vector(e1,ga=cnfml3d) # point a = (1,0,0) A = F(a)
B = make_vector(e2,ga=cnfml3d) # point b = (0,1,0) B = F(b)
C = make_vector(-e1,ga=cnfml3d) # point c = (-1,0,0) C = F(c)
D = make_vector(e3,ga=cnfml3d) # point d = (0,0,1) D = F(d)
X = make_vector('x',3,ga=cnfml3d)
print 'F(a) =',A
print 'F(b) =',B
print 'F(c) =',C
print 'F(d) =',D
print 'F(x) =',X
print 'a = e1, b = e2, c = -e1, and d = e3'
print 'A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.'
print 'Circle through a, b, and c'
print 'Circle: A^B^C^X = 0 =',(A^B^C^X)
print 'Line through a and b'
print 'Line : A^B^n^X = 0 =',(A^B^n^X)
print 'Sphere through a, b, c, and d'
print 'Sphere: A^B^C^D^X = 0 =',(((A^B)^C)^D)^X
print 'Plane through a, b, and d'
print 'Plane : A^B^n^D^X = 0 =',(A^B^n^D^X)
L = (A^B^e)^X
print L.Fmt(3,'Hyperbolic Circle: (A^B^e)^X = 0 =')
return
def test2_12_1_2(self):
"""
Given the 2-blade B = e_1 ^ (e_2 - e_3) that represents a plane,
determine if each of the following vectors lies in that plane.
"""
(_g3d, e_1, e_2, e_3) = Ga.build('e*1|2|3')
B = e_1 ^ (e_2 - e_3)
self.assertTrue(e_1 ^ B == 0)
self.assertFalse((e_1 + e_2) ^ B == 0)
self.assertFalse((e_1 + e_2 + e_3) ^ B == 0)
self.assertTrue((2*e_1 - e_2 + e_3) ^ B == 0)
def conformal_representations_of_circles_lines_spheres_and_planes():
Print_Function()
global n, nbar
Fmt(1)
g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'
c3d = Ga('e_1 e_2 e_3 n \\bar{n}', g=g)
(e1, e2, e3, n, nbar) = c3d.mv()
print('g_{ij} =', c3d.g)
e = n + nbar
#conformal representation of points
A = make_vector(e1, ga=c3d) # point a = (1,0,0) A = F(a)
B = make_vector(e2, ga=c3d) # point b = (0,1,0) B = F(b)
C = make_vector(-e1, ga=c3d) # point c = (-1,0,0) C = F(c)
D = make_vector(e3, ga=c3d) # point d = (0,0,1) D = F(d)
X = make_vector('x', 3, ga=c3d)
print('F(a) =', A)
print('F(b) =', B)
print('F(c) =', C)
print('F(d) =', D)
print('F(x) =', X)
print('#a = e1, b = e2, c = -e1, and d = e3')
print('#A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.')
print('#Circle through a, b, and c')
print('Circle: A^B^C^X = 0 =', (A ^ B ^ C ^ X))
print('#Line through a and b')
print('Line : A^B^n^X = 0 =', (A ^ B ^ n ^ X))
print('#Sphere through a, b, c, and d')
print('Sphere: A^B^C^D^X = 0 =', (((A ^ B) ^ C) ^ D) ^ X)
print('#Plane through a, b, and d')
print('Plane : A^B^n^D^X = 0 =', (A ^ B ^ n ^ D ^ X))
L = (A ^ B ^ e) ^ X
L.Fmt(3, 'Hyperbolic\\;\\; Circle: (A^B^e)^X = 0')
return
def test_2_12_1_1(self):
"""
Compute the outer products of the following 3-space expressions,
giving the result relative to the basis {1, e_1, e_2, e_3, e_1^e_2, e_1^e_3, e_2^e_3, e_1^e_2^e_3}.
"""
(_g3d, e_1, e_2, e_3) = Ga.build('e*1|2|3')
self.assertTrue((e_1 + e_2) ^ (e_1 + e_3) == (-e_1 ^ e_2) + (e_1 ^ e_3) + (e_2 ^ e_3))
self.assertTrue((e_1 + e_2 + e_3) ^ (2*e_1) == -2*(e_1 ^ e_2) - 2*(e_1 ^ e_3))
self.assertTrue((e_1 - e_2) ^ (e_1 - e_3) == (e_1 ^ e_2) - (e_1 ^ e_3) + (e_2 ^ e_3))
self.assertTrue((e_1 + e_2) ^ (0.5*e_1 + 2*e_2 + 3*e_3) == 1.5*(e_1 ^ e_2) + 3*(e_1 ^ e_3) + 3*(e_2 ^ e_3))
self.assertTrue((e_1 ^ e_2) ^ (e_1 + e_3) == (e_1 ^ e_2 ^ e_3))
self.assertTrue((e_1 + e_2) ^ ((e_1 ^ e_2) + (e_2 ^ e_3)) == (e_1 ^ e_2 ^ e_3))
def check_generalized_BAC_CAB_formulas():
Print_Function()
g5d = Ga('a b c d e')
(a, b, c, d, e) = g5d.mv()
print 'g_{ij} =\n', g5d.g
print 'a|(b*c) =', a | (b * c)
print 'a|(b^c) =', a | (b ^ c)
print 'a|(b^c^d) =', a | (b ^ c ^ d)
print 'a|(b^c)+c|(a^b)+b|(c^a) =', (a | ( b ^ c)) + (c | (a ^ b)) + (b | (c ^ a))
print 'a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b)
print '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 '(a^b)|(c^d) =',(a^b)|(c^d)
print '((a^b)|c)|d =',((a^b)|c)|d
print '(a^b)x(c^d) =',com(a^b,c^d)
print '(a|(b^c))|(d^e) =',(a|(b^c))|(d^e)
return
def Mv_setup_options():
Print_Function()
(o3d,e1,e2,e3) = Ga.build('e_1 e_2 e_3',g=[1,1,1])
v = o3d.mv('v', 'vector')
print v
(o3d,e1,e2,e3) = Ga.build('e*1|2|3',g=[1,1,1])
v = o3d.mv('v', 'vector')
print v
(o3d,e1,e2,e3) = Ga.build('e*x|y|z',g=[1,1,1])
v = o3d.mv('v', 'vector')
print v
coords = symbols('x y z',real=True)
(o3d,e1,e2,e3) = Ga.build('e',g=[1,1,1],coords=coords)
v = o3d.mv('v', 'vector')
print v
return
def check_generalized_BAC_CAB_formulas():
Print_Function()
g5d = Ga('a b c d e')
(a, b, c, d, e) = g5d.mv()
print('g_{ij} =\n', g5d.g)
print('a|(b*c) =', a | (b * c))
print('a|(b^c) =', a | (b ^ c))
print('a|(b^c^d) =', a | (b ^ c ^ d))
print('a|(b^c)+c|(a^b)+b|(c^a) =', (a | ( b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)))
print('a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b))
print('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('(a^b)|(c^d) =',(a^b)|(c^d))
print('((a^b)|c)|d =',((a^b)|c)|d)
print('(a^b)x(c^d) =',com(a^b,c^d))
print('(a|(b^c))|(d^e) =',(a|(b^c))|(d^e))
return
def Mv_setup_options():
Print_Function()
(o3d, e1, e2, e3) = Ga.build('e_1 e_2 e_3', g=[1, 1, 1])
v = o3d.mv('v', 'vector')
print v
(o3d, e1, e2, e3) = Ga.build('e*1|2|3', g=[1, 1, 1])
v = o3d.mv('v', 'vector')
print v
(o3d, e1, e2, e3) = Ga.build('e*x|y|z', g=[1, 1, 1])
v = o3d.mv('v', 'vector')
print v
coords = symbols('x y z', real=True)
(o3d, e1, e2, e3) = Ga.build('e', g=[1, 1, 1], coords=coords)
v = o3d.mv('v', 'vector')
print v
return
def derivatives_in_spherical_coordinates():
Print_Function()
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
def test_3_6(self):
"""
Orthogonal projection of subspaces.
"""
Ga.dual_mode("Iinv+")
R = Ga('e*1|2|3')
X_blades = [R.mv('X', i, 'grade') for i in range(R.n + 1)]
B_blades = [R.mv('B', i, 'grade') for i in range(R.n + 1)]
# projection of X on B
def P(X, B):
return (X < B.inv()) < B
# a projection should be idempotent
for X, B in product(X_blades, B_blades):
self.assertEquals(P(X, B), P(P(X, B), B))
# with the contraction
for X, B in product(X_blades, B_blades):
self.assertEquals(X < B, P(X, B) < B)
def basic_multivector_operations_2D_orthogonal():
Print_Function()
o2d = Ga('e*x|y',g=[1,1])
(ex,ey) = o2d.mv()
print 'g_{ii} =',o2d.g
X = o2d.mv('X','vector')
A = o2d.mv('A','spinor')
X.Fmt(1,'X')
A.Fmt(1,'A')
(X*A).Fmt(2,'X*A')
(X|A).Fmt(2,'X|A')
(X<A).Fmt(2,'X<A')
(X>A).Fmt(2,'X>A')
(A*X).Fmt(2,'A*X')
(A|X).Fmt(2,'A|X')
(A<X).Fmt(2,'A<X')
(A>X).Fmt(2,'A>X')
return
def basic_multivector_operations_2D_orthogonal():
Print_Function()
o2d = Ga('e*x|y',g=[1,1])
(ex,ey) = o2d.mv()
print 'g_{ii} =',o2d.g
X = o2d.mv('X','vector')
A = o2d.mv('A','spinor')
print X.Fmt(1,'X')
print A.Fmt(1,'A')
print (X*A).Fmt(2,'X*A')
print (X|A).Fmt(2,'X|A')
print (X<A).Fmt(2,'X<A')
print (X>A).Fmt(2,'X>A')
print (A*X).Fmt(2,'A*X')
print (A|X).Fmt(2,'A|X')
print (A<X).Fmt(2,'A<X')
print (A>X).Fmt(2,'A>X')
return
def coefs_test():
Print_Function()
(o3d, e1, e2, e3) = Ga.build("e_1 e_2 e_3", g=[1, 1, 1])
print o3d.blades_lst
print o3d.mv_blades_lst
v = o3d.mv("v", "vector")
print v
print v.blade_coefs([e3, e1])
A = o3d.mv("A", "mv")
print A
print A.blade_coefs([e1 ^ e3, e3, e1 ^ e2, e1 ^ e2 ^ e3])
print A.blade_coefs()
return
def coefs_test():
Print_Function()
(o3d, e1, e2, e3) = Ga.build('e_1 e_2 e_3', g=[1, 1, 1])
print(o3d.blades_lst)
print(o3d.mv_blades_lst)
v = o3d.mv('v', 'vector')
print(v)
print(v.blade_coefs([e3, e1]))
A = o3d.mv('A', 'mv')
print(A)
print(A.blade_coefs([e1 ^ e3, e3, e1 ^ e2, e1 ^ e2 ^ e3]))
print(A.blade_coefs())
return
def Lie_Group():
Print_Function()
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
a = st4d.mv('a','vector')
B = st4d.mv('B','bivector')
print('a =',a)
print('B =',B)
print('a|B =', a|B)
print(((a|B)|B).simplify().Fmt(3,'(a|B)|B'))
print((((a|B)|B)|B).simplify().Fmt(3,'((a|B)|B)|B'))
return
def main():
Format()
snr=1
g = '0 0 1 0 ,0 0 0 1 ,1 0 0 0 ,0 1 0 0'
sk4coords = (e1,e2,e3,e4) = symbols('e1 e2 e3 e4')
sk4 = Ga('e_1 e_2 e_3 e_4', g=g, coords=sk4coords)
(e1,e2,e3,e4) = sk4.mv()
print 'g_{ii} =',sk4.g
v = symbols('v', real=True)
x1=(e1+e3)/sqrt(2)
x2=(e2+e4)/sqrt(2)
print 'x_1<x_1==',x1<x1
print 'x_1<x_2==',x1<x2
print 'x_2<x_1==',x2<x1
print 'x_2<x_2==',x2<x2
print r'#$-\infty < v < \infty$'
print '(-v*(x_1^x_2)/2).exp()==',(-v*(x1^x2)/2).exp()
v = symbols('v', real=True, positive=True)
print r'#$0\le v < \infty$'
print '(-v*(x_1^x_2)/2).exp()==',(-v*(x1^x2)/2).exp()
xpdf()
return
def generateit():
#
# 繁衍多代
# 返回,最终编码coding,每次繁衍的编码decodings,每次繁衍的适应度值fitnessvalues,初始编码iniplot,每代的适应度值fitness
#
inipol = ga.codeit(n, length)
decodings = []
fitnessvalues = []
fitness = []
iniplot = np.copy(inipol).tolist()
for i in range(g):
inipol = multiply(inipol)[0]
decodings.append(multiply(inipol)[1])
fitnessvalues.append(multiply(inipol)[2])
fitness.append(np.mean(multiply(inipol)[2]))
return inipol, decodings, fitnessvalues, iniplot, fitness
def basic_multivector_operations_2D():
Print_Function()
(g2d,ex,ey) = Ga.build('e*x|y')
print('g_{ij} =',g2d.g)
X = g2d.mv('X','vector')
A = g2d.mv('A','spinor')
print(X.Fmt(1,'X'))
print(A.Fmt(1,'A'))
print((X|A).Fmt(2,'X|A'))
print((X<A).Fmt(2,'X<A'))
print((A>X).Fmt(2,'A>X'))
return
def test_check_generalized_BAC_CAB_formulas():
(a, b, c, d, e) = MV.setup('a b c d e')
assert str(a | (b * c)) == '-(a.c)*b + (a.b)*c'
assert str(a | (b ^ c)) == '-(a.c)*b + (a.b)*c'
assert str(a | (b ^ c ^ d)) == '(a.d)*b^c - (a.c)*b^d + (a.b)*c^d'
#assert str( (a|(b^c))+(c|(a^b))+(b|(c^a)) ) == '0'
assert str(a * (b ^ c) - b * (a ^ c) + c * (a ^ b)) == '3*a^b^c'
assert str(a * (b ^ c ^ d) - b * (a ^ c ^ d) + c * (a ^ b ^ d) - d *
(a ^ b ^ c)) == '4*a^b^c^d'
assert str((a ^ b) | (c ^ d)) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(((a ^ b) | c) | d) == '-(a.c)*(b.d) + (a.d)*(b.c)'
assert str(Ga.com(a ^ b, c
^ d)) == '-(b.d)*a^c + (b.c)*a^d + (a.d)*b^c - (a.c)*b^d'
assert str(
(a | (b ^ c)) | (d ^ e)
) == '(-(a.b)*(c.e) + (a.c)*(b.e))*d + ((a.b)*(c.d) - (a.c)*(b.d))*e'
return
def main():
Eprint()
(o3d, ex, ey, ez) = Ga.build('e*x|y|z', g=[1, 1, 1])
u = o3d.mv('u', 'vector')
v = o3d.mv('v', 'vector')
w = o3d.mv('w', 'vector')
print u
print v
uv = u ^ v
print uv
print uv.is_blade()
exp_uv = uv.exp()
print 'exp(uv) =', exp_uv
return
def EM_Waves_in_Geom_Calculus_Complex():
#Print_Function()
X = (t, x, y, z) = symbols('t x y z', real=True)
g = '1 # # 0,# 1 # 0,# # 1 0,0 0 0 -1'
coords = (xE, xB, xk, t) = symbols('x_E x_B x_k t', real=True)
(EBkst, eE, eB, ek, et) = Ga.build('e_E e_B e_k e_t', g=g, coords=coords)
i = EBkst.i
E, B, k, w = symbols('E B k omega', real=True)
F = E * eE * et + i * B * eB * et
K = k * ek + w * et
X = xE * eE + xB * eB + xk * ek + t * et
KX = (K | X).scalar()
F = F * exp(I * KX)
g = EBkst.g
print('g =', g)
print('X =', X)
print('K =', K)
print('K|X =', KX)
print('F =', F)
gradF = EBkst.grad * F
gradF = gradF.simplify()
(gradF).Fmt(3, 'grad*F = 0')
gradF = gradF.subs({g[0, 1]: 0, g[0, 2]: 0, g[1, 2]: 0})
KX = KX.subs({g[0, 1]: 0, g[0, 2]: 0, g[1, 2]: 0})
print(
r'%\mbox{Substituting }e_{E}\cdot e_{B} = e_{E}\cdot e_{k} = e_{B}\cdot e_{k} = 0'
)
(gradF / (I * exp(I * KX))).Fmt(
3, r'%\lp\bm{\nabla}F\rp/\lp ie^{iK\cdot X}\rp = 0')
return
def test_is_base(self):
"""
Various tests on several multivectors.
"""
(_g3d, e_1, e_2, e_3) = Ga.build('e*1|2|3')
self.assertTrue((e_1).is_base())
self.assertTrue((e_2).is_base())
self.assertTrue((e_3).is_base())
self.assertTrue((e_1 ^ e_2).is_base())
self.assertTrue((e_2 ^ e_3).is_base())
self.assertTrue((e_1 ^ e_3).is_base())
self.assertTrue((e_1 ^ e_2 ^ e_3).is_base())
self.assertFalse((2 * e_1).is_base())
self.assertFalse((e_1 + e_2).is_base())
self.assertFalse((e_3 * 4).is_base())
self.assertFalse(((3 * e_1) ^ e_2).is_base())
self.assertFalse((2 * (e_2 ^ e_3)).is_base())
self.assertFalse((e_3 ^ e_1).is_base())
self.assertFalse((e_2 ^ e_1 ^ e_3).is_base())