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 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', fct=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) =', -MV.I * (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 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 main():
enhance_print()
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, y, z))
A = x*(ey ^ ez) + y*(ez ^ ex) + z*(ex ^ ey)
print('A =', A)
print('grad^A =', (grad ^ A).simplify())
print()
f = MV('f', 'scalar', fct=True)
f = (x**2 + y**2 + z**2)**(-1.5)
print('f =', f)
print('grad*f =', (grad*f).expand())
print()
B = f*A
print('B =', B)
print()
Curl_B = grad ^ B
print('grad^B =', Curl_B.simplify())
def Symplify(A):
return(factor_terms(simplify(A)))
print(Curl_B.func(Symplify))
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 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 Test_Reciprocal_Frame():
Print_Function()
Format()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e_x e_y e_z', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
eu = ex + ey
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('\\mbox{Frame}', (eu, ev), '\\mbox{Reciprocal Frame}', (eu_r, ev_r))
print(r'%\bm{e}_{u}\cdot\bm{e}^{u} =', (eu | eu_r))
print(r'%\bm{e}_{u}\cdot\bm{e}^{v} =', eu | ev_r)
print(r'%\bm{e}_{v}\cdot\bm{e}^{u} =', ev | eu_r)
print(r'%\bm{e}_{v}\cdot\bm{e}^{v} =', ev | ev_r)
eu = ex + ey + ez
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('\\mbox{Frame}', (eu, ev), '\\mbox{Reciprocal Frame}', (eu_r, ev_r))
print(r'%\bm{e}_{u}\cdot\bm{e}^{u} =', eu | eu_r)
print(r'%\bm{e}_{u}\cdot\bm{e}^{v} =', eu | ev_r)
print(r'%\bm{e}_{v}\cdot\bm{e}^{u} =', ev | eu_r)
print(r'%\bm{e}_{v}\cdot\bm{e}^{v} =', ev | ev_r)
return
def Simple_manifold_with_scalar_function_derivative():
Print_Function()
coords = (x, y, z) = symbols('x y z')
basis = (e1, e2, e3, grad) = MV.setup('e_1 e_2 e_3',
metric='[1,1,1]',
coords=coords)
# Define surface
mfvar = (u, v) = symbols('u v')
X = u * e1 + v * e2 + (u**2 + v**2) * e3
print('\\f{X}{u,v} =', X)
MF = Manifold(X, mfvar)
(eu, ev) = MF.Basis()
# Define field on the surface.
g = (v + 1) * log(u)
print('\\f{g}{u,v} =', g)
# Method 1: Using old Manifold routines.
VectorDerivative = (MF.rbasis[0] / MF.E_sq) * diff(
g, u) + (MF.rbasis[1] / MF.E_sq) * diff(g, v)
print('\\eval{\\nabla g}{u=1,v=0} =', VectorDerivative.subs({u: 1, v: 0}))
# Method 2: Using new Manifold routines.
dg = MF.Grad(g)
print('\\eval{\\f{Grad}{g}}{u=1,v=0} =', dg.subs({u: 1, v: 0}))
dg = MF.grad * g
print('\\eval{\\nabla g}{u=1,v=0} =', dg.subs({u: 1, v: 0}))
return
def Distorted_manifold_with_scalar_function():
Print_Function()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e_x e_y e_z',
metric='[1,1,1]',
coords=coords)
mfvar = (u, v) = symbols('u v')
X = 2 * u * ex + 2 * v * ey + (u**3 + v**3 / 2) * ez
MF = Manifold(X, mfvar, I=MV.I)
(eu, ev) = MF.Basis()
g = (v + 1) * log(u)
dg = MF.Grad(g)
print('g =', g)
print('dg =', dg)
print('\\eval{dg}{u=1,v=0} =', dg.subs({u: 1, v: 0}))
G = u * eu + v * ev
dG = MF.Grad(G)
print('G =', G)
print('P(G) =', MF.Proj(G))
print('dG =', dG)
print('P(dG) =', MF.Proj(dG))
PS = u * v * eu ^ ev
print('P(S) =', PS)
print('dP(S) =', MF.Grad(PS))
print('P(dP(S)) =', MF.Proj(MF.Grad(PS)))
return
def Simple_manifold_with_vector_function_derivative():
Print_Function()
coords = (x, y, z) = symbols('x y z')
basis = (ex, ey, ez, grad) = \
MV.setup('e_x e_y e_z', metric='[1,1,1]', coords=coords)
# Define surface
mfvar = (u, v) = symbols('u v')
X = u * ex + v * ey + (u**2 + v**2) * ez
print('\\f{X}{u,v} =', X)
MF = Manifold(X, mfvar)
(eu, ev) = MF.Basis()
# Define field on the surface.
g = (v + 1) * log(u)
print('\\mbox{Scalar Function: } g =', g)
dg = MF.grad * g
dg.Fmt(3, '\\mbox{Scalar Function Derivative: } \\nabla g')
print('\\eval{\\nabla g}{(1,0)} =', dg.subs({u: 1, v: 0}))
# Define vector field on the surface
G = v**2 * eu + u**2 * ev
print('\\mbox{Vector Function: } G =', G)
dG = MF.grad * G
dG.Fmt(3, '\\mbox{Vector Function Derivative: } \\nabla G')
print('\\eval{\\nabla G}{(1,0)} =', dG.subs({u: 1, v: 0}))
return
def Test_Reciprocal_Frame():
Print_Function()
Format()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e_x e_y e_z',
metric='[1,1,1]',
coords=coords)
mfvar = (u, v) = symbols('u v')
eu = ex + ey
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('\\mbox{Frame}', (eu, ev), '\\mbox{Reciprocal Frame}', (eu_r, ev_r))
print(r'%\bm{e}_{u}\cdot\bm{e}^{u} =', (eu | eu_r))
print(r'%\bm{e}_{u}\cdot\bm{e}^{v} =', eu | ev_r)
print(r'%\bm{e}_{v}\cdot\bm{e}^{u} =', ev | eu_r)
print(r'%\bm{e}_{v}\cdot\bm{e}^{v} =', ev | ev_r)
eu = ex + ey + ez
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('\\mbox{Frame}', (eu, ev), '\\mbox{Reciprocal Frame}', (eu_r, ev_r))
print(r'%\bm{e}_{u}\cdot\bm{e}^{u} =', eu | eu_r)
print(r'%\bm{e}_{u}\cdot\bm{e}^{v} =', eu | ev_r)
print(r'%\bm{e}_{v}\cdot\bm{e}^{u} =', ev | eu_r)
print(r'%\bm{e}_{v}\cdot\bm{e}^{v} =', ev | ev_r)
return
def properties_of_geometric_objects():
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 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 Simple_manifold_with_vector_function_derivative():
Print_Function()
coords = (x, y, z) = symbols('x y z')
basis = (ex, ey, ez, grad) = \
MV.setup('e_x e_y e_z', metric='[1,1,1]', coords=coords)
# Define surface
mfvar = (u, v) = symbols('u v')
X = u*ex + v*ey + (u**2 + v**2)*ez
print('\\f{X}{u,v} =', X)
MF = Manifold(X, mfvar)
(eu, ev) = MF.Basis()
# Define field on the surface.
g = (v + 1)*log(u)
print('\\mbox{Scalar Function: } g =', g)
dg = MF.grad*g
dg.Fmt(3, '\\mbox{Scalar Function Derivative: } \\nabla g')
print('\\eval{\\nabla g}{(1,0)} =', dg.subs({u: 1, v: 0}))
# Define vector field on the surface
G = v**2*eu + u**2*ev
print('\\mbox{Vector Function: } G =', G)
dG = MF.grad*G
dG.Fmt(3, '\\mbox{Vector Function Derivative: } \\nabla G')
print('\\eval{\\nabla G}{(1,0)} =', dG.subs({u: 1, v: 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([xp, tp])
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(gamma))
return
def extracting_vectors_from_conformal_2_blade():
Print_Function()
metric = " 0 -1 #," + "-1 0 #," + " # # #,"
(P1, P2, a) = MV.setup("P1 P2 a", metric)
print("g_{ij} =\n", 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 properties_of_geometric_objects():
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 nbar', metric)
print('g_{ij} =\n', MV.metric)
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 Distorted_manifold_with_scalar_function():
Print_Function()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e_x e_y e_z', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
X = 2*u*ex + 2*v*ey + (u**3 + v**3/2)*ez
MF = Manifold(X, mfvar, I=MV.I)
(eu, ev) = MF.Basis()
g = (v + 1)*log(u)
dg = MF.Grad(g)
print('g =', g)
print('dg =', dg)
print('\\eval{dg}{u=1,v=0} =', dg.subs({u: 1, v: 0}))
G = u*eu + v*ev
dG = MF.Grad(G)
print('G =', G)
print('P(G) =', MF.Proj(G))
print('dG =', dG)
print('P(dG) =', MF.Proj(dG))
PS = u*v*eu ^ ev
print('P(S) =', PS)
print('dP(S) =', MF.Grad(PS))
print('P(dP(S)) =', MF.Proj(MF.Grad(PS)))
return
def check_generalized_BAC_CAB_formulas():
Print_Function()
(a, b, c, d, e) = MV.setup('a b c d e')
print('g_{ij} =\n', MV.metric)
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 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', fct=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) =', -MV.I*(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 derivatives_in_spherical_coordinates():
Print_Function()
X = (r, th, phi) = symbols('r theta phi')
curv = [[r *
cos(phi) *
sin(th), r *
sin(phi) *
sin(th), r *
cos(th)], [1, r, r *
sin(th)]]
(er,
eth,
ephi,
grad) = MV.setup('e_r e_theta e_phi',
metric='[1,1,1]',
coords=X,
curv=curv)
f = MV('f', 'scalar', fct=True)
A = MV('A', 'vector', fct=True)
B = MV('B', 'grade2', fct=True)
print('f =', f)
print('A =', A)
print('B =', B)
print('grad*f =', grad*f)
print('grad|A =', grad | A)
print('-I*(grad^A) =', -MV.I*(grad ^ A))
print('grad^B =', grad ^ B)
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([xp, tp])
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(gamma))
return
def reciprocal_frame_test():
Print_Function()
metric = "1 # #," + "# 1 #," + "# # 1,"
(e1, e2, e3) = MV.setup("e1 e2 e3", metric)
print("g_{ij} =\n", MV.metric)
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)
w = E1 | e2
w = w.expand()
print("E1|e2 =", w)
w = E1 | e3
w = w.expand()
print("E1|e3 =", w)
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|e1)/E**2 =", simplify(w / Esq))
w = E2 | e2
w = (w.expand()).scalar()
print("(E2|e2)/E**2 =", simplify(w / Esq))
w = E3 | e3
w = (w.expand()).scalar()
print("(E3|e3)/E**2 =", simplify(w / Esq))
return
def extracting_vectors_from_conformal_2_blade():
Print_Function()
metric = ' 0 -1 #,' + \
'-1 0 #,' + \
' # # #,'
(P1, P2, a) = MV.setup('P1 P2 a', metric)
print('g_{ij} =\n', 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 properties_of_geometric_objects():
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 nbar", metric)
print("g_{ij} =\n", MV.metric)
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 Simple_manifold_with_scalar_function_derivative():
Print_Function()
coords = (x, y, z) = symbols('x y z')
basis = (e1, e2, e3, grad) = MV.setup('e_1 e_2 e_3', metric='[1,1,1]', coords=coords)
# Define surface
mfvar = (u, v) = symbols('u v')
X = u*e1 + v*e2 + (u**2 + v**2)*e3
print('\\f{X}{u,v} =', X)
MF = Manifold(X, mfvar)
(eu, ev) = MF.Basis()
# Define field on the surface.
g = (v + 1)*log(u)
print('\\f{g}{u,v} =', g)
# Method 1: Using old Manifold routines.
VectorDerivative = (MF.rbasis[0]/MF.E_sq)*diff(g, u) + (MF.rbasis[1]/MF.E_sq)*diff(g, v)
print('\\eval{\\nabla g}{u=1,v=0} =', VectorDerivative.subs({u: 1, v: 0}))
# Method 2: Using new Manifold routines.
dg = MF.Grad(g)
print('\\eval{\\f{Grad}{g}}{u=1,v=0} =', dg.subs({u: 1, v: 0}))
dg = MF.grad*g
print('\\eval{\\nabla g}{u=1,v=0} =', dg.subs({u: 1, v: 0}))
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", fct=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) =", -MV.I * (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 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 main():
enhance_print()
(ex, ey, ez) = MV.setup('e*x|y|z', metric='[1,1,1]')
u = MV('u', 'vector')
v = MV('v', 'vector')
w = MV('w', 'vector')
print(u)
print(v)
uv = u ^ v
print(uv)
print(uv.is_blade())
exp_uv = uv.exp()
exp_uv.Fmt(2, 'exp(uv)')
return
def Plot_Mobius_Strip_Manifold():
Print_Function()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e_x e_y e_z', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
X = (cos(u) + v*cos(u/2)*cos(u))*ex + (sin(u) + v*cos(u/2)*sin(u))*ey + v*sin(u/2)*ez
MF = Manifold(X, mfvar, True, I=MV.I)
MF.Plot2DSurface([0.0, 6.28, 48], [-0.3, 0.3, 12], surf=False, skip=[4, 4], tan=0.15)
return
def make_vector(a, n=3):
if isinstance(a, str):
sym_str = ''
for i in range(n):
sym_str += a + str(i + 1) + ' '
sym_lst = list(symbols(sym_str))
sym_lst.append(S.Zero)
sym_lst.append(S.Zero)
a = MV(sym_lst, 'vector')
return (F(a))
def derivatives_in_spherical_coordinates():
Print_Function()
X = (r, th, phi) = symbols('r theta phi')
curv = [[r*cos(phi)*sin(th), r*sin(phi)*sin(th), r*cos(th)], [1, r, r*sin(th)]]
(er, eth, ephi, grad) = MV.setup('e_r e_theta e_phi', metric='[1,1,1]', coords=X, curv=curv)
f = MV('f', 'scalar', fct=True)
A = MV('A', 'vector', fct=True)
B = MV('B', 'grade2', fct=True)
print('f =', f)
print('A =', A)
print('B =', B)
print('grad*f =', grad*f)
print('grad|A =', grad | A)
print('-I*(grad^A) =', -MV.I*(grad ^ A))
print('grad^B =', grad ^ B)
return
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()
(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 MV_setup_options():
Print_Function()
(e1, e2, e3) = MV.setup('e_1 e_2 e_3', '[1,1,1]')
v = MV('v', 'vector')
print(v)
(e1, e2, e3) = MV.setup('e*1|2|3', '[1,1,1]')
v = MV('v', 'vector')
print(v)
(e1, e2, e3) = MV.setup('e*x|y|z', '[1,1,1]')
v = MV('v', 'vector')
print(v)
coords = symbols('x y z')
(e1, e2, e3, grad) = MV.setup('e', '[1,1,1]', coords=coords)
v = MV('v', 'vector')
print(v)
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 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 Plot_Mobius_Strip_Manifold():
Print_Function()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e_x e_y e_z',
metric='[1,1,1]',
coords=coords)
mfvar = (u, v) = symbols('u v')
X = (cos(u) + v * cos(u / 2) * cos(u)) * ex + (
sin(u) + v * cos(u / 2) * sin(u)) * ey + v * sin(u / 2) * ez
MF = Manifold(X, mfvar, True, I=MV.I)
MF.Plot2DSurface([0.0, 6.28, 48], [-0.3, 0.3, 12],
surf=False,
skip=[4, 4],
tan=0.15)
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)} =', 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 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 check_generalized_BAC_CAB_formulas():
Print_Function()
(a, b, c, d, e) = MV.setup('a b c d e')
print('g_{ij} =\n', MV.metric)
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()
(e1, e2, e3) = MV.setup('e_1 e_2 e_3', '[1,1,1]')
v = MV('v', 'vector')
print(v)
(e1, e2, e3) = MV.setup('e*1|2|3', '[1,1,1]')
v = MV('v', 'vector')
print(v)
(e1, e2, e3) = MV.setup('e*x|y|z', '[1,1,1]')
v = MV('v', 'vector')
print(v)
coords = symbols('x y z')
(e1, e2, e3, grad) = MV.setup('e', '[1,1,1]', coords=coords)
v = MV('v', 'vector')
print(v)
return
def derivatives_in_spherical_coordinates():
Print_Function()
X = (r, th, phi) = symbols("r theta phi")
curv = [[r * cos(phi) * sin(th), r * sin(phi) * sin(th), r * cos(th)], [1, r, r * sin(th)]]
(er, eth, ephi, grad) = MV.setup("e_r e_theta e_phi", metric="[1,1,1]", coords=X, curv=curv)
f = MV("f", "scalar", fct=True)
A = MV("A", "vector", fct=True)
B = MV("B", "grade2", fct=True)
print("f =", f)
print("A =", A)
print("B =", B)
print("grad*f =", grad * f)
print("grad|A =", grad | A)
print("-I*(grad^A) =", -MV.I * (grad ^ A))
print("grad^B =", grad ^ B)
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)} =', Com(a ^ b, c ^ d))
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 conformal_representations_of_circles_lines_spheres_and_planes():
global n, nbar
Print_Function()
metric = '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'
(e1, e2, e3, n, nbar) = MV.setup('e_1 e_2 e_3 n nbar', metric)
print('g_{ij} =\n', MV.metric)
e = n + nbar
# conformal representation of points
A = make_vector(e1) # point a = (1,0,0) A = F(a)
B = make_vector(e2) # point b = (0,1,0) B = F(b)
C = make_vector(-e1) # point c = (-1,0,0) C = F(c)
D = make_vector(e3) # point d = (0,0,1) D = F(d)
X = make_vector('x', 3)
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 conformal_representations_of_circles_lines_spheres_and_planes():
global n, nbar
Print_Function()
metric = '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'
(e1, e2, e3, n, nbar) = MV.setup('e_1 e_2 e_3 n nbar', metric)
print('g_{ij} =\n', MV.metric)
e = n + nbar
#conformal representation of points
A = make_vector(e1) # point a = (1,0,0) A = F(a)
B = make_vector(e2) # point b = (0,1,0) B = F(b)
C = make_vector(-e1) # point c = (-1,0,0) C = F(c)
D = make_vector(e3) # point d = (0,0,1) D = F(d)
X = make_vector('x', 3)
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 main():
enhance_print()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('ex ey ez', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
eu = ex + ey
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('Frame', (eu, ev), 'Reciprocal Frame', (eu_r, ev_r))
print('eu.eu_r =', eu | eu_r)
print('eu.ev_r =', eu | ev_r)
print('ev.eu_r =', ev | eu_r)
print('ev.ev_r =', ev | ev_r)
eu = ex + ey + ez
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('Frame', (eu, ev), 'Reciprocal Frame', (eu_r, ev_r))
print('eu.eu_r =', eu | eu_r)
print('eu.ev_r =', eu | ev_r)
print('ev.eu_r =', ev | eu_r)
print('ev.ev_r =', ev | ev_r)
print('eu =', eu)
print('ev =', ev)
define_precedence(locals())
print(GAeval('eu^ev|ex', True))
print(GAeval('eu^ev|ex*eu', True))
return
def conformal_representations_of_circles_lines_spheres_and_planes():
global n, nbar
Print_Function()
metric = "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"
(e1, e2, e3, n, nbar) = MV.setup("e_1 e_2 e_3 n nbar", metric)
print("g_{ij} =\n", MV.metric)
e = n + nbar
# conformal representation of points
A = make_vector(e1) # point a = (1,0,0) A = F(a)
B = make_vector(e2) # point b = (0,1,0) B = F(b)
C = make_vector(-e1) # point c = (-1,0,0) C = F(c)
D = make_vector(e3) # point d = (0,0,1) D = F(d)
X = make_vector("x", 3)
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 basic_multivector_operations_2D_orthogonal():
Print_Function()
(ex, ey) = MV.setup('e*x|y', metric='[1,1]')
print('g_{ii} =', 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 main():
Format()
(ex, ey, ez) = MV.setup('e*x|y|z')
A = 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')
(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)
print(r'\bm{A} =', A)
print(r'\bm{B} =', B)
print('grad*f =', grad * f)
print(r'grad|\bm{A} =', grad | A)
print(r'grad*\bm{A} =', grad * A)
print(r'-I*(grad^\bm{A}) =', -MV.I * (grad ^ A))
print(r'grad*\bm{B} =', grad * B)
print(r'grad^\bm{B} =', grad ^ B)
print(r'grad|\bm{B} =', grad | B)
(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)} =', Com(a ^ b, c ^ d))
metric = '1 # #,' + \
'# 1 #,' + \
'# # 1,'
(e1, e2, e3) = MV.setup('e1 e2 e3', metric)
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')
curv = [[r * cos(phi) * sin(th), r * sin(phi) * sin(th), r * cos(th)],
[1, r, r * sin(th)]]
(er, eth, ephi, grad) = MV.setup('e_r e_theta e_phi',
metric='[1,1,1]',
coords=X,
curv=curv)
f = MV('f', 'scalar', fct=True)
A = MV('A', 'vector', fct=True)
B = MV('B', 'grade2', fct=True)
print('A =', A)
print('B =', B)
print('grad*f =', grad * f)
print('grad|A =', grad | A)
print('-I*(grad^A) =', -MV.I * (grad ^ A))
print('grad^B =', grad ^ B)
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
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('B = \\bm{B\\gamma_{t}} =', B)
print('E = \\bm{E\\gamma_{t}} =', E)
print('F = E+IB =', F)
print('J =', J)
gradF = grad * F
gradF.Fmt(3, 'grad*F')
print('grad*F = J')
(gradF.grade(1) - J).Fmt(3, '%\\grade{\\nabla F}_{1} -J = 0')
(gradF.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'")
(g0, g1) = MV.setup('gamma*t|x', metric='[1,-1]')
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([xp, tp])
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(gamma))
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('\\bm{A} =', A)
print('\\bm{\\psi} =', psi)
dirac_eq = (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'
)
xdvi()
return
def reciprocal_frame_test():
Print_Function()
metric = '1 # #,' + \
'# 1 #,' + \
'# # 1,'
(e1, e2, e3) = MV.setup('e1 e2 e3', metric)
print('g_{ij} =\n', MV.metric)
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)
w = (E1 | e2)
w = w.expand()
print('E1|e2 =', w)
w = (E1 | e3)
w = w.expand()
print('E1|e3 =', w)
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|e1)/E**2 =', simplify(w / Esq))
w = (E2 | e2)
w = (w.expand()).scalar()
print('(E2|e2)/E**2 =', simplify(w / Esq))
w = (E3 | e3)
w = (w.expand()).scalar()
print('(E3|e3)/E**2 =', simplify(w / Esq))
return
def noneuclidian_distance_calculation():
Print_Function()
from sympy import solve, sqrt
metric = '0 # #,# 0 #,# # 1'
(X, Y, e) = MV.setup('X Y e', metric)
print('g_{ij} =', MV.metric)
print('%(X\\W Y)^{2} =', (X ^ Y) * (X ^ Y))
L = X ^ Y ^ e
B = L * e # D&L 10.152
Bsq = (B * B).scalar()
print('#%L = X\\W Y\\W e \\text{ is a non-euclidian line}')
print('B = L*e =', B)
BeBr = B * e * B.rev()
print('%BeB^{\\dagger} =', BeBr)
print('%B^{2} =', B * B)
print('%L^{2} =', L * L) # D&L 10.153
(s, c, Binv, M, BigS, BigC, alpha, XdotY, Xdote,
Ydote) = symbols('s c (1/B) M S C alpha (X.Y) (X.e) (Y.e)')
Bhat = Binv * B # D&L 10.154
R = c + s * Bhat # Rotor R = exp(alpha*Bhat/2)
print(
'#%s = \\f{\\sinh}{\\alpha/2} \\text{ and } c = \\f{\\cosh}{\\alpha/2}'
)
print('%e^{\\alpha B/{2\\abs{B}}} =', R)
Z = R * X * R.rev() # D&L 10.155
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv, s, c, XdotY])
Z.Fmt(3, '%RXR^{\\dagger}')
W = Z | Y # Extract scalar part of multivector
# From this point forward all calculations are with sympy scalars
# print '#Objective is to determine value of C = cosh(alpha) such that W = 0'
W = W.scalar()
print('%W = Z\\cdot Y =', W)
W = expand(W)
W = simplify(W)
W = W.collect([s * Binv])
M = 1 / Bsq
W = W.subs(Binv**2, M)
W = simplify(W)
Bmag = sqrt(XdotY**2 - 2 * XdotY * Xdote * Ydote)
W = W.collect([Binv * c * s, XdotY])
# Double angle substitutions
W = W.subs(2 * XdotY**2 - 4 * XdotY * Xdote * Ydote, 2 / (Binv**2))
W = W.subs(2 * c * s, BigS)
W = W.subs(c**2, (BigC + 1) / 2)
W = W.subs(s**2, (BigC - 1) / 2)
W = simplify(W)
W = W.subs(1 / Binv, Bmag)
W = expand(W)
print('#%S = \\f{\\sinh}{\\alpha} \\text{ and } C = \\f{\\cosh}{\\alpha}')
print('W =', W)
Wd = collect(W, [BigC, BigS], exact=True, evaluate=False)
Wd_1 = Wd[S.One]
Wd_C = Wd[BigC]
Wd_S = Wd[BigS]
print('%\\text{Scalar Coefficient} =', Wd_1)
print('%\\text{Cosh Coefficient} =', Wd_C)
print('%\\text{Sinh Coefficient} =', Wd_S)
print('%\\abs{B} =', Bmag)
Wd_1 = Wd_1.subs(Bmag, 1 / Binv)
Wd_C = Wd_C.subs(Bmag, 1 / Binv)
Wd_S = Wd_S.subs(Bmag, 1 / Binv)
lhs = Wd_1 + Wd_C * BigC
rhs = -Wd_S * BigS
lhs = lhs**2
rhs = rhs**2
W = expand(lhs - rhs)
W = expand(W.subs(1 / Binv**2, Bmag**2))
W = expand(W.subs(BigS**2, BigC**2 - 1))
W = W.collect([BigC, BigC**2], evaluate=False)
a = simplify(W[BigC**2])
b = simplify(W[BigC])
c = simplify(W[S.One])
print('#%\\text{Require } aC^{2}+bC+c = 0')
print('a =', a)
print('b =', b)
print('c =', c)
x = Symbol('x')
C = solve(a * x**2 + b * x + c, x)[0]
print('%b^{2}-4ac =', simplify(b**2 - 4 * a * c))
print('%\\f{\\cosh}{\\alpha} = C = -b/(2a) =', expand(simplify(expand(C))))
return
def noneuclidian_distance_calculation():
from sympy import solve, sqrt
Print_Function()
metric = '0 # #,# 0 #,# # 1'
(X, Y, e) = MV.setup('X Y e', metric)
print('g_{ij} =', MV.metric)
print('(X^Y)**2 =', (X ^ Y) * (X ^ Y))
L = X ^ Y ^ e
B = L * e # D&L 10.152
print('B =', B)
Bsq = B * B
print('B**2 =', Bsq)
Bsq = Bsq.scalar()
print('#L = X^Y^e is a non-euclidian line')
print('B = L*e =', B)
BeBr = B * e * B.rev()
print('B*e*B.rev() =', BeBr)
print('B**2 =', B * B)
print('L**2 =', L * L) # D&L 10.153
(s, c, Binv, M, BigS, BigC, alpha, XdotY, Xdote,
Ydote) = symbols('s c (1/B) M S C alpha (X.Y) (X.e) (Y.e)')
Bhat = Binv * B # D&L 10.154
R = c + s * Bhat # Rotor R = exp(alpha*Bhat/2)
print('s = sinh(alpha/2) and c = cosh(alpha/2)')
print('exp(alpha*B/(2*|B|)) =', R)
Z = R * X * R.rev() # D&L 10.155
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv, s, c, XdotY])
Z.Fmt(3, 'R*X*R.rev()')
W = Z | Y # Extract scalar part of multivector
# From this point forward all calculations are with sympy scalars
print('Objective is to determine value of C = cosh(alpha) such that W = 0')
W = W.scalar()
print('Z|Y =', W)
W = expand(W)
W = simplify(W)
W = W.collect([s * Binv])
M = 1 / Bsq
W = W.subs(Binv**2, M)
W = simplify(W)
Bmag = sqrt(XdotY**2 - 2 * XdotY * Xdote * Ydote)
W = W.collect([Binv * c * s, XdotY])
#Double angle substitutions
W = W.subs(2 * XdotY**2 - 4 * XdotY * Xdote * Ydote, 2 / (Binv**2))
W = W.subs(2 * c * s, BigS)
W = W.subs(c**2, (BigC + 1) / 2)
W = W.subs(s**2, (BigC - 1) / 2)
W = simplify(W)
W = W.subs(1 / Binv, Bmag)
W = expand(W)
print('S = sinh(alpha) and C = cosh(alpha)')
print('W =', W)
Wd = collect(W, [BigC, BigS], exact=True, evaluate=False)
Wd_1 = Wd[S.One]
Wd_C = Wd[BigC]
Wd_S = Wd[BigS]
print('Scalar Coefficient =', Wd_1)
print('Cosh Coefficient =', Wd_C)
print('Sinh Coefficient =', Wd_S)
print('|B| =', Bmag)
Wd_1 = Wd_1.subs(Bmag, 1 / Binv)
Wd_C = Wd_C.subs(Bmag, 1 / Binv)
Wd_S = Wd_S.subs(Bmag, 1 / Binv)
lhs = Wd_1 + Wd_C * BigC
rhs = -Wd_S * BigS
lhs = lhs**2
rhs = rhs**2
W = expand(lhs - rhs)
W = expand(W.subs(1 / Binv**2, Bmag**2))
W = expand(W.subs(BigS**2, BigC**2 - 1))
W = W.collect([BigC, BigC**2], evaluate=False)
a = simplify(W[BigC**2])
b = simplify(W[BigC])
c = simplify(W[S.One])
print('Require a*C**2+b*C+c = 0')
print('a =', a)
print('b =', b)
print('c =', c)
x = Symbol('x')
C = solve(a * x**2 + b * x + c, x)[0]
print('cosh(alpha) = C = -b/(2*a) =', expand(simplify(expand(C))))
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 main():
Format()
coords = (x, y, z) = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('e*x|y|z', '[1,1,1]', coords=coords)
s = MV('s', 'scalar')
v = MV('v', 'vector')
b = MV('b', 'bivector')
print(r'#3D Orthogonal Metric\newline')
print('#Multvectors:')
print('s =', s)
print('v =', v)
print('b =', b)
print('#Products:')
X = ((s, 's'), (v, 'v'), (b, 'b'))
for xi in X:
print('')
for yi in X:
print(xi[1] + '*' + yi[1] + ' =', xi[0] * yi[0])
print(xi[1] + '^' + yi[1] + ' =', xi[0] ^ yi[0])
print(xi[1] + '|' + yi[1] + ' =', xi[0] | yi[0])
print(xi[1] + '<' + yi[1] + ' =', xi[0] < yi[0])
print(xi[1] + '>' + yi[1] + ' =', xi[0] > yi[0])
fs = MV('s', 'scalar', fct=True)
fv = MV('v', 'vector', fct=True)
fb = MV('b', 'bivector', fct=True)
print('#Multivector Functions:')
print('s(X) =', fs)
print('v(X) =', fv)
print('b(X) =', fb)
print('#Products:')
fX = ((grad, 'grad'), (fs, 's'), (fv, 'v'), (fb, 'b'))
for xi in fX:
print('')
for yi in fX:
if xi[1] == 'grad' and yi[1] == 'grad':
pass
else:
print(xi[1] + '*' + yi[1] + ' =', xi[0] * yi[0])
print(xi[1] + '^' + yi[1] + ' =', xi[0] ^ yi[0])
print(xi[1] + '|' + yi[1] + ' =', xi[0] | yi[0])
print(xi[1] + '<' + yi[1] + ' =', xi[0] < yi[0])
print(xi[1] + '>' + yi[1] + ' =', xi[0] > yi[0])
(ex, ey, grad) = MV.setup('e', coords=(x, y))
print(r'#General 2D Metric\newline')
print('#Multivector Functions:')
s = MV('s', 'scalar', fct=True)
v = MV('v', 'vector', fct=True)
b = MV('v', 'bivector', fct=True)
print('s(X) =', s)
print('v(X) =', v)
print('b(X) =', b)
X = ((grad, 'grad'), (s, 's'), (v, 'v'))
print('#Products:')
for xi in X:
print('')
for yi in X:
if xi[1] == 'grad' and yi[1] == 'grad':
pass
else:
print(xi[1] + '*' + yi[1] + ' =', xi[0] * yi[0])
print(xi[1] + '^' + yi[1] + ' =', xi[0] ^ yi[0])
print(xi[1] + '|' + yi[1] + ' =', xi[0] | yi[0])
print(xi[1] + '<' + yi[1] + ' =', xi[0] < yi[0])
print(xi[1] + '>' + yi[1] + ' =', xi[0] > yi[0])
xdvi(paper='letter')
return
def basic_multivector_operations_3D():
Print_Function()
(ex, ey, ez) = MV.setup('e*x|y|z')
print('g_{ij} =', MV.metric)
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')
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
