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()
(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 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 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 basic_multivector_operations_2D():
(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 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 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)).simplify())
print('grad^B =',grad^B)
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 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(ZERO)
sym_lst.append(ZERO)
a = MV(sym_lst,'vector')
return(F(a))
def MV_setup_options():
(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():
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():
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)
print(w)
uv = u ^ v
print(uv)
print(uv.is_blade())
uvw = u ^ v ^ w
print(uvw)
print(uvw.is_blade())
print(simplify((uv * uv).scalar()))
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
from __future__ import print_function
from sympy import *
from galgebra.ga import Ga
from galgebra.deprecated import MV
from galgebra.printer import Format, xpdf, Fmt
Format()
ew, ex, ey, ez = MV.setup('e_w e_x e_y e_z', metric=[1, 1, 1, 1])
a = MV('a', 'vector')
a.set_coef(1, 0, 0)
b = ex + ey + ez
c = MV('c', 'vector')
print('a =', a)
print('b =', b)
print('c =', c)
print(a.reflect_in_blade(ex ^ ey).Fmt(1, 'a\\mbox{ reflect in }xy'))
print(a.reflect_in_blade(ey ^ ez).Fmt(1, 'a\\mbox{ reflect in }yz'))
print(a.reflect_in_blade(ez ^ ex).Fmt(1, 'a\\mbox{ reflect in }zx'))
print(
a.reflect_in_blade(ez ^ (ex + ey)).Fmt(1,
'a\\mbox{ reflect in plane }(x=y)'))
print(
b.reflect_in_blade((ez - ey) ^ (ex - ey)).Fmt(
1, 'b\\mbox{ reflect in plane }(x+y+z=0)'))
print(a.reflect_in_blade(ex).Fmt(1, '\\mbox{Reflect in }\\bm{e}_{x}'))
print(a.reflect_in_blade(ey).Fmt(1, '\\mbox{Reflect in }\\bm{e}_{y}'))
print(a.reflect_in_blade(ez).Fmt(1, '\\mbox{Reflect in }\\bm{e}_{z}'))
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])
if xi[1] != 's' and yi[1] != 's':
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])
if xi[1] != 's' and yi[1] != 's':
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])
if xi[1] != 's' and yi[1] != 's':
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 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)} =', Ga.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()
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())
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'
)
# xpdf()
xpdf(pdfprog=None)
return