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 extracting_vectors_from_conformal_2_blade():
global n,nbar
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():
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 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 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 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 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 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 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 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_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 test_extracting_vectors_from_conformal_2_blade():
metric = ' 0 -1 #,'+ \
'-1 0 #,'+ \
' # # #,'
(P1, P2, a) = MV.setup('P1 P2 a', metric)
B = P1 ^ P2
Bsq = B * B
assert str(Bsq) == '1'
ap = a - (a ^ B) * B
assert str(ap) == '-(P2.a)*P1 - (P1.a)*P2'
Ap = ap + ap * B
Am = ap - ap * B
assert str(Ap) == '-2*(P2.a)*P1'
assert str(Am) == '-2*(P1.a)*P2'
assert str(Ap * Ap) == '0'
assert str(Am * Am) == '0'
aB = a | B
assert str(aB) == '-(P2.a)*P1 + (P1.a)*P2'
return
def test_derivatives_in_spherical_coordinates():
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)
assert str(f) == 'f'
assert str(A) == 'A__r*e_r + A__theta*e_theta + A__phi*e_phi'
assert str(
B
) == 'B__rtheta*e_r^e_theta + B__rphi*e_r^e_phi + B__thetaphi*e_theta^e_phi'
assert str(
grad *
f) == 'D{r}f*e_r + D{theta}f/r*e_theta + D{phi}f/(r*sin(theta))*e_phi'
assert str(
grad | A
) == 'D{r}A__r + 2*A__r/r + A__theta*cos(theta)/(r*sin(theta)) + D{theta}A__theta/r + D{phi}A__phi/(r*sin(theta))'
assert str(
-MV.I * (grad ^ A)
) == '((A__phi*cos(theta)/sin(theta) + D{theta}A__phi - D{phi}A__theta/sin(theta))/r)*e_r + (-D{r}A__phi - A__phi/r + D{phi}A__r/(r*sin(theta)))*e_theta + (D{r}A__theta + A__theta/r - D{theta}A__r/r)*e_phi'
assert str(
grad ^ B
) == '(D{r}B__thetaphi - B__rphi*cos(theta)/(r*sin(theta)) + 2*B__thetaphi/r - D{theta}B__rphi/r + D{phi}B__rtheta/(r*sin(theta)))*e_r^e_theta^e_phi'
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 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 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 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 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_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
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 test_rounding_numerical_components():
(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
#assert str(X) == '1.20000000000000*e_x + 2.34000000000000*e_y + 0.555000000000000*e_z'
assert str(Nga(X, 2)) == '1.2*e_x + 2.3*e_y + 0.55*e_z'
#assert str(X*Y) == '12.7011000000000 + 4.02078000000000*e_x^e_y + 6.17518500000000*e_x^e_z + 10.1820000000000*e_y^e_z'
assert str(Nga(X * Y,
2)) == '13. + 4.0*e_x^e_y + 6.2*e_x^e_z + 10.*e_y^e_z'
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 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 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 test_derivatives_in_rectangular_coordinates():
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)
assert str(f) == 'f'
assert str(A) == 'A__x*e_x + A__y*e_y + A__z*e_z'
assert str(B) == 'B__xy*e_x^e_y + B__xz*e_x^e_z + B__yz*e_y^e_z'
assert str(
C
) == 'C + C__x*e_x + C__y*e_y + C__z*e_z + C__xy*e_x^e_y + C__xz*e_x^e_z + C__yz*e_y^e_z + C__xyz*e_x^e_y^e_z'
assert str(grad * f) == 'D{x}f*e_x + D{y}f*e_y + D{z}f*e_z'
assert str(grad | A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(
grad * A
) == 'D{x}A__x + D{y}A__y + D{z}A__z + (-D{y}A__x + D{x}A__y)*e_x^e_y + (-D{z}A__x + D{x}A__z)*e_x^e_z + (-D{z}A__y + D{y}A__z)*e_y^e_z'
assert str(
-MV.I * (grad ^ A)
) == '(-D{z}A__y + D{y}A__z)*e_x + (D{z}A__x - D{x}A__z)*e_y + (-D{y}A__x + D{x}A__y)*e_z'
assert str(
grad * B
) == '(-D{y}B__xy - D{z}B__xz)*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z + (D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z'
assert str(grad ^ B) == '(D{z}B__xy - D{y}B__xz + D{x}B__yz)*e_x^e_y^e_z'
assert str(
grad | B
) == '(-D{y}B__xy - D{z}B__xz)*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z'
assert str(grad < A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(grad > A) == 'D{x}A__x + D{y}A__y + D{z}A__z'
assert str(
grad < B
) == '(-D{y}B__xy - D{z}B__xz)*e_x + (D{x}B__xy - D{z}B__yz)*e_y + (D{x}B__xz + D{y}B__yz)*e_z'
assert str(grad > B) == '0'
assert str(
grad < C
) == 'D{x}C__x + D{y}C__y + D{z}C__z + (-D{y}C__xy - D{z}C__xz)*e_x + (D{x}C__xy - D{z}C__yz)*e_y + (D{x}C__xz + D{y}C__yz)*e_z + D{z}C__xyz*e_x^e_y - D{y}C__xyz*e_x^e_z + D{x}C__xyz*e_y^e_z'
assert str(
grad > C
) == 'D{x}C__x + D{y}C__y + D{z}C__z + D{x}C*e_x + D{y}C*e_y + D{z}C*e_z'
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():
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 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 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 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 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 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_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 test_conformal_representations_of_circles_lines_spheres_and_planes():
global n, nbar
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)
e = n + nbar
#conformal representation of points
A = make_vector(e1)
B = make_vector(e2)
C = make_vector(-e1)
D = make_vector(e3)
X = make_vector('x', 3)
assert str(A) == 'e_1 + 1/2*n - 1/2*nbar'
assert str(B) == 'e_2 + 1/2*n - 1/2*nbar'
assert str(C) == '-e_1 + 1/2*n - 1/2*nbar'
assert str(D) == 'e_3 + 1/2*n - 1/2*nbar'
assert str(
X
) == 'x1*e_1 + x2*e_2 + x3*e_3 + ((x1**2 + x2**2 + x3**2)/2)*n - 1/2*nbar'
assert str(
(A ^ B ^ C ^ X)
) == '-x3*e_1^e_2^e_3^n + x3*e_1^e_2^e_3^nbar + ((x1**2 + x2**2 + x3**2 - 1)/2)*e_1^e_2^n^nbar'
assert str(
(A ^ B ^ n ^ X)
) == '-x3*e_1^e_2^e_3^n + ((x1 + x2 - 1)/2)*e_1^e_2^n^nbar + x3/2*e_1^e_3^n^nbar - x3/2*e_2^e_3^n^nbar'
assert str((((A ^ B) ^ C) ^ D)
^ X) == '((-x1**2 - x2**2 - x3**2 + 1)/2)*e_1^e_2^e_3^n^nbar'
assert str(
(A ^ B ^ n ^ D ^ X)) == '((-x1 - x2 - x3 + 1)/2)*e_1^e_2^e_3^n^nbar'
L = (A ^ B ^ e) ^ X
assert str(
L
) == '-x3*e_1^e_2^e_3^n - x3*e_1^e_2^e_3^nbar + (-x1**2/2 + x1 - x2**2/2 + x2 - x3**2/2 - 1/2)*e_1^e_2^n^nbar + x3*e_1^e_3^n^nbar - x3*e_2^e_3^n^nbar'
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 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 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
def_prec(locals())
print GAeval('eu^ev|ex',True)
print GAeval('eu^ev|ex*eu',True)
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)
def_prec(locals())
print(GAeval('eu^ev|ex', True))
print(GAeval('eu^ev|ex*eu', True))
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 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 test_properties_of_geometric_objects():
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)
P1 = F(p1)
P2 = F(p2)
P3 = F(p3)
L = P1 ^ P2 ^ n
delta = (L | n) | nbar
assert str(delta) == '2*p1 - 2*p2'
C = P1 ^ P2 ^ P3
delta = ((C ^ n) | n) | nbar
assert str(delta) == '2*p1^p2 - 2*p1^p3 + 2*p2^p3'
assert str((p2 - p1) ^ (p3 - p1)) == 'p1^p2 - p1^p3 + p2^p3'
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
from sympy import *
from ga import Ga
from mv import MV
from 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}')
print c.reflect_in_blade(ex^ey).Fmt(1,'c\\mbox{ reflect in }xy')
print c.reflect_in_blade(ex^ey^ez).Fmt(1,'c\\mbox{ reflect in }xyz')
print (ew^ex).reflect_in_blade(ey^ez).Fmt(1,'wx\\mbox{ reflect in }yz')
print (ew^ex).reflect_in_blade(ex^ey).Fmt(1,'wx\\mbox{ reflect in }xy')
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,S,C,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,S)
W = W.subs(c**2,(C+1)/2)
W = W.subs(s**2,(C-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,[C,S],exact=True,evaluate=False)
Wd_1 = Wd[ONE]
Wd_C = Wd[C]
Wd_S = Wd[S]
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*C
rhs = -Wd_S*S
lhs = lhs**2
rhs = rhs**2
W = expand(lhs-rhs)
W = expand(W.subs(1/Binv**2,Bmag**2))
W = expand(W.subs(S**2,C**2-1))
W = W.collect([C,C**2],evaluate=False)
a = simplify(W[C**2])
b = simplify(W[C])
c = simplify(W[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 reciprocal_frame_test():
Print_Function()
metric = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
(e1,e2,e3) = MV.setup('e1 e2 e3',metric)
print 'g_{ij} =',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\\cdot e1)/E^{2} =',simplify(w/Esq)
w = (E2|e2)
w = (w.expand()).scalar()
print '%(E2\\cdot e2)/E^{2} =',simplify(w/Esq)
w = (E3|e3)
w = (w.expand()).scalar()
print '%(E3\\cdot e3)/E^{2} =',simplify(w/Esq)
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,S,C,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,S)
W = W.subs(c**2,(C+1)/2)
W = W.subs(s**2,(C-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,[C,S],exact=True,evaluate=False)
Wd_1 = Wd[ONE]
Wd_C = Wd[C]
Wd_S = Wd[S]
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*C
rhs = -Wd_S*S
lhs = lhs**2
rhs = rhs**2
W = expand(lhs-rhs)
W = expand(W.subs(1/Binv**2,Bmag**2))
W = expand(W.subs(S**2,C**2-1))
W = W.collect([C,C**2],evaluate=False)
a = simplify(W[C**2])
b = simplify(W[C])
c = simplify(W[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 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
from sympy import S
from printer import xdvi,Get_Program,Print_Function,Format
from mv import MV
(ex,ey,ez) = MV.setup('e_x e_y e_z',metric='[1,1,1]')
v = S(3)*ex+S(4)*ey
print v
print v.norm()
print v.norm2()
print v/v.norm()
print v/(v.norm()**2)
print v*(v/v.norm2())
print v.inv()
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()
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')
xdvi()
return