def derivatives_in_toroidal_coordinates():
Print_Function()
a = symbols('a', real=True)
coords = (u, v, phi) = symbols('u v phi', real=True)
(t3d, eu, ev,
ephi) = Ga.build('e_u e_v e_phi',
X=[
a * sinh(v) * cos(phi) / (cosh(v) - cos(u)),
a * sinh(v) * sin(phi) / (cosh(v) - cos(u)),
a * sin(u) / (cosh(v) - cos(u))
],
coords=coords,
norm=True)
grad = t3d.grad
f = t3d.mv('f', 'scalar', f=True)
A = t3d.mv('A', 'vector', f=True)
B = t3d.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) =', -t3d.i * (grad ^ A))
print('grad^B =', grad ^ B)
return
def properties_of_geometric_objects():
global n,nbar
Print_Function()
metric = '# # # 0 0,'+ \
'# # # 0 0,'+ \
'# # # 0 0,'+ \
'0 0 0 0 2,'+ \
'0 0 0 2 0'
(p1,p2,p3,n,nbar) = MV.setup('p1 p2 p3 n \\bar{n}',metric)
print('g_{ij} =',MV.metric)
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 derivatives_in_oblate_spheroidal_coordinates():
Print_Function()
a = symbols('a', real=True)
coords = (xi, eta, phi) = symbols('xi eta phi', real=True)
(os3d, er, eth, ephi) = Ga.build('e_xi e_eta e_phi',
X=[
a * cosh(xi) * cos(eta) * cos(phi),
a * cosh(xi) * cos(eta) * sin(phi),
a * sinh(xi) * sin(eta)
],
coords=coords,
norm=True)
grad = os3d.grad
f = os3d.mv('f', 'scalar', f=True)
A = os3d.mv('A', 'vector', f=True)
B = os3d.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) =', -os3d.i * (grad ^ A))
print('grad^B =', grad ^ B)
return
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 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 derivatives_in_rectangular_coordinates():
Print_Function()
X = (x,y,z) = symbols('x y z')
(ex,ey,ez,grad) = MV.setup('e_x e_y e_z',metric='[1,1,1]',coords=X)
f = MV('f','scalar',fct=True)
A = MV('A','vector',fct=True)
B = MV('B','grade2',fct=True)
C = 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(-MV.I)
print('-I*(grad^A) =',-MV.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,ex,ey,ez) = Ga.build('e*x|y|z')
print('g_{ij} =',g3d.g)
A = g3d.mv('A','mv')
print(A.Fmt(1,'A'))
print(A.Fmt(2,'A'))
print(A.Fmt(3,'A'))
print(A.even().Fmt(1,'%A_{+}'))
print(A.odd().Fmt(1,'%A_{-}'))
X = g3d.mv('X','vector')
Y = g3d.mv('Y','vector')
print(X.Fmt(1,'X'))
print(Y.Fmt(1,'Y'))
print((X*Y).Fmt(2,'X*Y'))
print((X^Y).Fmt(2,'X^Y'))
print((X|Y).Fmt(2,'X|Y'))
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 extracting_vectors_from_conformal_2_blade():
Print_Function()
print(r'B = P1\W P2')
metric = '0 -1 #,'+ \
'-1 0 #,'+ \
'# # #'
(P1,P2,a) = MV.setup('P1 P2 a',metric)
print('g_{ij} =',MV.metric)
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 Dirac_Equation_in_Geometric_Calculus():
Print_Function()
vars = symbols('t x y z')
(g0, g1, g2, g3, grad) = MV.setup('gamma*t|x|y|z',
metric='[1,-1,-1,-1]',
coords=vars)
I = MV.I
(m, e) = symbols('m e')
psi = MV('psi', 'spinor', fct=True)
A = MV('A', 'vector', fct=True)
sig_z = g3 * g0
print('\\text{4-Vector Potential\\;\\;}\\bm{A} =', A)
print('\\text{8-component real spinor\\;\\;}\\bm{\\psi} =', psi)
dirac_eq = (grad * psi) * I * sig_z - e * A * psi - m * psi * g0
dirac_eq.simplify()
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 extracting_vectors_from_conformal_2_blade():
Print_Function()
g = '0 -1 #,'+ \
'-1 0 #,'+ \
'# # #'
e2b = Ga('P1 P2 a', g=g)
(P1, P2, a) = e2b.mv()
print('g_{ij} =\n', e2b.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 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 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 properties_of_geometric_objects():
Print_Function()
global n, nbar
Fmt(1)
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 basic_multivector_operations_3D():
Print_Function()
g3d = Ga('e*x|y|z')
(ex, ey, ez) = g3d.mv()
A = g3d.mv('A', 'mv')
print(A.Fmt(1, 'A'))
print(A.Fmt(2, 'A'))
print(A.Fmt(3, 'A'))
print(A.even().Fmt(1, '%A_{+}'))
print(A.odd().Fmt(1, '%A_{-}'))
X = g3d.mv('X', 'vector')
Y = g3d.mv('Y', 'vector')
print('g_{ij} = ', g3d.g)
print(X.Fmt(1, 'X'))
print(Y.Fmt(1, 'Y'))
print((X * Y).Fmt(2, 'X*Y'))
print((X ^ Y).Fmt(2, 'X^Y'))
print((X | Y).Fmt(2, 'X|Y'))
print(cross(X, Y).Fmt(1, r'X\times Y'))
return
def basic_multivector_operations_3D():
Print_Function()
(ex,ey,ez) = MV.setup('e*x|y|z')
A = 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 = MV('X','vector')
Y = MV('Y','vector')
print('g_{ij} = ',MV.metric)
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 Lorentz_Tranformation_in_Geometric_Algebra():
Print_Function()
(alpha,beta,gamma) = symbols('alpha beta gamma')
(x,t,xp,tp) = symbols("x t x' t'")
(g0,g1) = MV.setup('gamma*t|x',metric='[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.subs({2*sinh(alpha/2)*cosh(alpha/2):sinh(alpha),sinh(alpha/2)**2+cosh(alpha/2)**2:cosh(alpha)})
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 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 nbar', g=g)
(p1, p2, p3, n, nbar) = c3d.mv()
print('g_{ij} =\n', c3d.g)
P1 = F(p1)
P2 = F(p2)
P3 = F(p3)
print('Extracting direction of line from L = P1^P2^n')
L = P1 ^ P2 ^ n
delta = (L | n) | nbar
print('(L|n)|nbar =', delta)
print('Extracting plane of circle from C = P1^P2^P3')
C = P1 ^ P2 ^ P3
delta = ((C ^ n) | n) | nbar
print('((C^n)|n)|nbar =', delta)
print('(p2-p1)^(p3-p1) =', (p2 - p1) ^ (p3 - p1))
def Maxwells_Equations_in_Geometric_Calculus():
Print_Function()
X = symbols('t x y z')
(g0,g1,g2,g3,grad) = MV.setup('gamma*t|x|y|z',metric='[1,-1,-1,-1]',coords=X)
I = MV.I
B = MV('B','vector',fct=True)
E = MV('E','vector',fct=True)
B.set_coef(1,0,0)
E.set_coef(1,0,0)
B *= g0
E *= g0
J = MV('J','vector',fct=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 = grad*F
print('#Geometric 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.grade(1)-J).Fmt(3,'%\\grade{\\nabla F}_{1} -J = 0')
print('#Curl $E$ and Div $B$ equations')
(gradF.grade(3)).Fmt(3,'%\\grade{\\nabla F}_{3} = 0')
return
def signature_test():
Print_Function()
e3d = Ga('e1 e2 e3', g=[1, 1, 1])
print('e3d.g =', e3d.g)
print('Signature = (3,0) I =', e3d.I(), ' I**2 =', e3d.I() * e3d.I())
e3d = Ga('e1 e2 e3', g=[2, 2, 2])
print('e3d.g =', e3d.g)
print('Signature = (3,0) I =', e3d.I(), ' I**2 =', e3d.I() * e3d.I())
sp4d = Ga('e1 e2 e3 e4', g=[1, -1, -1, -1])
print('e3d.g =', sp4d.g)
print('Signature = (1,3) I =', sp4d.I(), ' I**2 =', sp4d.I() * sp4d.I())
sp4d = Ga('e1 e2 e3 e4', g=[2, -2, -2, -2])
print('e3d.g =', sp4d.g)
print('Signature = (1,3) I =', sp4d.I(), ' I**2 =', sp4d.I() * sp4d.I())
e4d = Ga('e1 e2 e3 e4', g=[1, 1, 1, 1])
print('e4d.g =', e4d.g)
print('Signature = (4,0) I =', e4d.I(), ' I**2 =', e4d.I() * e4d.I())
cf3d = Ga('e1 e2 e3 e4 e5', g=[1, 1, 1, 1, -1])
print('cf4d.g =', cf3d.g)
print('Signature = (4,1) I =', cf3d.I(), ' I**2 =', cf3d.I() * cf3d.I())
cf3d = Ga('e1 e2 e3 e4 e5', g=[2, 2, 2, 2, -2])
print('cf4d.g =', cf3d.g)
print('Signature = (4,1) I =', cf3d.I(), ' I**2 =', cf3d.I() * cf3d.I())
return
def derivatives_in_bipolar_coordinates():
Print_Function()
a = symbols('a', real=True)
coords = (u, v, z) = symbols('u v z', real=True)
(bp3d, eu, ev, ez) = Ga.build('e_u e_v e_z',
X=[
a * sinh(v) / (cosh(v) - cos(u)),
a * sin(u) / (cosh(v) - cos(u)), z
],
coords=coords,
norm=True)
grad = bp3d.grad
f = bp3d.mv('f', 'scalar', f=True)
A = bp3d.mv('A', 'vector', f=True)
B = bp3d.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) =', -bp3d.i * (grad ^ A))
print('grad^B =', grad ^ B)
return
def basic_multivector_operations():
Print_Function()
(ex, ey, ez) = MV.setup('e*x|y|z')
A = MV('A', 'mv')
A.Fmt(1, 'A')
A.Fmt(2, 'A')
A.Fmt(3, 'A')
X = MV('X', 'vector')
Y = MV('Y', 'vector')
print('g_{ij} =\n', MV.metric)
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')
(ex, ey) = MV.setup('e*x|y')
print('g_{ij} =\n', MV.metric)
X = MV('X', 'vector')
A = 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')
(ex, ey) = MV.setup('e*x|y', metric='[1,1]')
print('g_{ii} =\n', MV.metric)
X = MV('X', 'vector')
A = 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 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 rounding_numerical_components():
Print_Function()
(ex,ey,ez) = MV.setup('e_x e_y e_z',metric='[1,1,1]')
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 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 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 check_generalized_BAC_CAB_formulas():
Print_Function()
(a,b,c,d) = MV.setup('a b c d')
print('g_{ij} =',MV.metric)
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)} =',Ga.com(a^b,c^d))
return
def basic_multivector_operations_2D():
Print_Function()
(ex,ey) = MV.setup('e*x|y')
print('g_{ij} =',MV.metric)
X = MV('X','vector')
A = 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 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
