def test_derivatives_in_spherical_coordinates(self):
X = (r, th, phi) = symbols('r theta phi')
s3d = Ga('e_r e_theta e_phi', g=[1, r ** 2, r ** 2 * sin(th) ** 2], coords=X, norm=True)
(er, eth, ephi) = s3d.mv()
grad = s3d.grad
f = s3d.mv('f', 'scalar', f=True)
A = s3d.mv('A', 'vector', f=True)
B = s3d.mv('B', 'bivector', f=True)
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*e_theta/r + D{phi}f*e_phi/(r*sin(theta))'
assert str((grad|A).simplify()) == '(r*D{r}A__r + 2*A__r + A__theta/tan(theta) + D{theta}A__theta + D{phi}A__phi/sin(theta))/r'
assert str(-s3d.I()*(grad^A)) == '(A__phi/tan(theta) + D{theta}A__phi - D{phi}A__theta/sin(theta))*e_r/r + (-r*D{r}A__phi - A__phi + D{phi}A__r/sin(theta))*e_theta/r + (r*D{r}A__theta + A__theta - D{theta}A__r)*e_phi/r'
assert latex(grad) == r'\boldsymbol{e}_{r} \frac{\partial}{\partial r} + \boldsymbol{e}_{\theta } \frac{1}{r} \frac{\partial}{\partial \theta } + \boldsymbol{e}_{\phi } \frac{1}{r \sin{\left (\theta \right )}} \frac{\partial}{\partial \phi }'
assert latex(B|(eth^ephi)) == r'- B^{\theta \phi } {\left (r,\theta ,\phi \right )}'
assert str(grad^B) == '(r*D{r}B__thetaphi - B__rphi/tan(theta) + 2*B__thetaphi - D{theta}B__rphi + D{phi}B__rtheta/sin(theta))*e_r^e_theta^e_phi/r'
return
def test_derivatives_in_rectangular_coordinates(self):
X = (x, y, z) = symbols('x y z')
o3d = Ga('e_x e_y e_z', g=[1, 1, 1], coords=X)
(ex, ey, ez) = o3d.mv()
grad = o3d.grad
f = o3d.mv('f', 'scalar', f=True)
A = o3d.mv('A', 'vector', f=True)
B = o3d.mv('B', 'bivector', f=True)
C = o3d.mv('C', 'mv', f=True)
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(-o3d.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_derivatives_in_rectangular_coordinates(self):
X = (x, y, z) = symbols('x y z')
o3d = Ga('e_x e_y e_z', g=[1, 1, 1], coords=X)
(ex, ey, ez) = o3d.mv()
grad = o3d.grad
f = o3d.mv('f', 'scalar', f=True)
A = o3d.mv('A', 'vector', f=True)
B = o3d.mv('B', 'bivector', f=True)
C = o3d.mv('C', 'mv', f=True)
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(
-o3d.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) == ZERO_STR
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'
def test_extracting_vectors_from_conformal_2_blade(self):
g = '0 -1 #,'+ \
'-1 0 #,'+ \
'# # #'
e2b = Ga('P1 P2 a',g=g)
(P1,P2,a) = e2b.mv()
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_properties_of_geometric_objects(self):
global n, nbar
g = '# # # 0 0,'+ \
'# # # 0 0,'+ \
'# # # 0 0,'+ \
'0 0 0 0 2,'+ \
'0 0 0 2 0'
c3d = Ga('p1 p2 p3 n nbar',g=g)
(p1,p2,p3,n,nbar) = c3d.mv()
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 test_conformal_representations_of_circles_lines_spheres_and_planes(self):
global n,nbar
g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'
cnfml3d = Ga('e_1 e_2 e_3 n nbar',g=g)
(e1,e2,e3,n,nbar) = cnfml3d.mv()
e = n+nbar
#conformal representation of points
A = make_vector(e1,ga=cnfml3d) # point a = (1,0,0) A = F(a)
B = make_vector(e2,ga=cnfml3d) # point b = (0,1,0) B = F(b)
C = make_vector(-e1,ga=cnfml3d) # point c = (-1,0,0) C = F(c)
D = make_vector(e3,ga=cnfml3d) # point d = (0,0,1) D = F(d)
X = make_vector('x',3,ga=cnfml3d)
assert str(A) == 'e_1 + n/2 - nbar/2'
assert str(B) == 'e_2 + n/2 - nbar/2'
assert str(C) == '-e_1 + n/2 - nbar/2'
assert str(D) == 'e_3 + n/2 - nbar/2'
assert str(X) == 'x1*e_1 + x2*e_2 + x3*e_3 + (x1**2/2 + x2**2/2 + x3**2/2)*n - nbar/2'
assert str((A^B^C^X)) == '-x3*e_1^e_2^e_3^n + x3*e_1^e_2^e_3^nbar + (x1**2/2 + x2**2/2 + x3**2/2 - 1/2)*e_1^e_2^n^nbar'
assert str((A^B^n^X)) == '-x3*e_1^e_2^e_3^n + (x1/2 + x2/2 - 1/2)*e_1^e_2^n^nbar + x3*e_1^e_3^n^nbar/2 - x3*e_2^e_3^n^nbar/2'
assert str((((A^B)^C)^D)^X) == '(-x1**2/2 - x2**2/2 - x3**2/2 + 1/2)*e_1^e_2^e_3^n^nbar'
assert str((A^B^n^D^X)) == '(-x1/2 - x2/2 - x3/2 + 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 test_properties_of_geometric_objects(self):
global n, nbar
g = '# # # 0 0,'+ \
'# # # 0 0,'+ \
'# # # 0 0,'+ \
'0 0 0 0 2,'+ \
'0 0 0 2 0'
c3d = Ga('p1 p2 p3 n nbar', g=g)
(p1, p2, p3, n, nbar) = c3d.mv()
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'
def test_extracting_vectors_from_conformal_2_blade(self):
g = '0 -1 #,'+ \
'-1 0 #,'+ \
'# # #'
e2b = Ga('P1 P2 a', g=g)
(P1, P2, a) = e2b.mv()
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) == ZERO_STR
assert str(Am * Am) == ZERO_STR
aB = a | B
assert str(aB) == '-(P2.a)*P1 + (P1.a)*P2'
def properties_of_geometric_objects():
Print_Function()
global n, nbar
Fmt(1)
g = '# # # 0 0,'+ \
'# # # 0 0,'+ \
'# # # 0 0,'+ \
'0 0 0 0 2,'+ \
'0 0 0 2 0'
c3d = Ga('p1 p2 p3 n \\bar{n}', g=g)
(p1, p2, p3, n, nbar) = c3d.mv()
print('g_{ij} =', c3d.g)
P1 = F(p1)
P2 = F(p2)
P3 = F(p3)
print('\\text{Extracting direction of line from }L = P1\\W P2\\W n')
L = P1 ^ P2 ^ n
delta = (L | n) | nbar
print('(L|n)|\\bar{n} =', delta)
print('\\text{Extracting plane of circle from }C = P1\\W P2\\W P3')
C = P1 ^ P2 ^ P3
delta = ((C ^ n) | n) | nbar
print('((C^n)|n)|\\bar{n}=', delta)
print('(p2-p1)^(p3-p1)=', (p2 - p1) ^ (p3 - p1))
return
def properties_of_geometric_objects():
Print_Function()
global n, nbar
g = '# # # 0 0,'+ \
'# # # 0 0,'+ \
'# # # 0 0,'+ \
'0 0 0 0 2,'+ \
'0 0 0 2 0'
c3d = Ga('p1 p2 p3 n nbar', g=g)
(p1, p2, p3, n, nbar) = c3d.mv()
print('g_{ij} =\n', c3d.g)
P1 = F(p1)
P2 = F(p2)
P3 = F(p3)
print('Extracting direction of line from L = P1^P2^n')
L = P1 ^ P2 ^ n
delta = (L | n) | nbar
print('(L|n)|nbar =', delta)
print('Extracting plane of circle from C = P1^P2^P3')
C = P1 ^ P2 ^ P3
delta = ((C ^ n) | n) | nbar
print('((C^n)|n)|nbar =', delta)
print('(p2-p1)^(p3-p1) =', (p2 - p1) ^ (p3 - p1))
def extracting_vectors_from_conformal_2_blade():
Print_Function()
g = '0 -1 #,'+ \
'-1 0 #,'+ \
'# # #'
e2b = Ga('P1 P2 a', g=g)
(P1, P2, a) = e2b.mv()
print('g_{ij} =\n', e2b.g)
B = P1 ^ P2
Bsq = B * B
print('B**2 =', Bsq)
ap = a - (a ^ B) * B
print("a' = a-(a^B)*B =", ap)
Ap = ap + ap * B
Am = ap - ap * B
print("A+ = a'+a'*B =", Ap)
print("A- = a'-a'*B =", Am)
print('(A+)^2 =', Ap * Ap)
print('(A-)^2 =', Am * Am)
aB = a | B
print('a|B =', aB)
return
def test_conformal_representations_of_circles_lines_spheres_and_planes(self):
global n,nbar
g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'
cnfml3d = Ga('e_1 e_2 e_3 n nbar',g=g)
(e1,e2,e3,n,nbar) = cnfml3d.mv()
e = n+nbar
#conformal representation of points
A = make_vector(e1,ga=cnfml3d) # point a = (1,0,0) A = F(a)
B = make_vector(e2,ga=cnfml3d) # point b = (0,1,0) B = F(b)
C = make_vector(-e1,ga=cnfml3d) # point c = (-1,0,0) C = F(c)
D = make_vector(e3,ga=cnfml3d) # point d = (0,0,1) D = F(d)
X = make_vector('x',3,ga=cnfml3d)
assert str(A) == 'e_1 + n/2 - nbar/2'
assert str(B) == 'e_2 + n/2 - nbar/2'
assert str(C) == '-e_1 + n/2 - nbar/2'
assert str(D) == 'e_3 + n/2 - nbar/2'
assert str(X) == 'x1*e_1 + x2*e_2 + x3*e_3 + (x1**2/2 + x2**2/2 + x3**2/2)*n - nbar/2'
assert str((A^B^C^X)) == '-x3*e_1^e_2^e_3^n + x3*e_1^e_2^e_3^nbar + (x1**2/2 + x2**2/2 + x3**2/2 - 1/2)*e_1^e_2^n^nbar'
assert str((A^B^n^X)) == '-x3*e_1^e_2^e_3^n + (x1/2 + x2/2 - 1/2)*e_1^e_2^n^nbar + x3*e_1^e_3^n^nbar/2 - x3*e_2^e_3^n^nbar/2'
assert str((((A^B)^C)^D)^X) == '(-x1**2/2 - x2**2/2 - x3**2/2 + 1/2)*e_1^e_2^e_3^n^nbar'
assert str((A^B^n^D^X)) == '(-x1/2 - x2/2 - x3/2 + 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 basic_multivector_operations_2D():
Print_Function()
g2d = Ga('e*x|y')
(ex, ey) = g2d.mv()
print('g_{ij} =', g2d.g)
X = g2d.mv('X', 'vector')
A = g2d.mv('A', 'spinor')
print(X.Fmt(1, 'X'))
print(A.Fmt(1, 'A'))
print((X | A).Fmt(2, 'X|A'))
print((X < A).Fmt(2, 'X<A'))
print((A > X).Fmt(2, 'A>X'))
return
def test_derivatives_in_spherical_coordinates(self):
X = (r, th, phi) = symbols('r theta phi')
s3d = Ga('e_r e_theta e_phi',
g=[1, r**2, r**2 * sin(th)**2],
coords=X,
norm=True)
(er, eth, ephi) = s3d.mv()
grad = s3d.grad
f = s3d.mv('f', 'scalar', f=True)
A = s3d.mv('A', 'vector', f=True)
B = s3d.mv('B', 'bivector', f=True)
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*e_theta/r + D{phi}f*e_phi/(r*sin(theta))'
assert str(
(grad | A).simplify()
) == '(r*D{r}A__r + 2*A__r + A__theta/tan(theta) + D{theta}A__theta + D{phi}A__phi/sin(theta))/r'
assert str(
-s3d.I() * (grad ^ A)
) == '(A__phi/tan(theta) + D{theta}A__phi - D{phi}A__theta/sin(theta))*e_r/r + (-r*D{r}A__phi - A__phi + D{phi}A__r/sin(theta))*e_theta/r + (r*D{r}A__theta + A__theta - D{theta}A__r)*e_phi/r'
assert latex(
grad
) == r'\boldsymbol{e}_{r} \frac{\partial}{\partial r} + \boldsymbol{e}_{\theta } \frac{1}{r} \frac{\partial}{\partial \theta } + \boldsymbol{e}_{\phi } \frac{1}{r \sin{\left (\theta \right )}} \frac{\partial}{\partial \phi }'
assert latex(B | (
eth
^ ephi)) == r'- B^{\theta \phi } {\left (r,\theta ,\phi \right )}'
assert str(
grad ^ B
) == '(r*D{r}B__thetaphi - B__rphi/tan(theta) + 2*B__thetaphi - D{theta}B__rphi + D{phi}B__rtheta/sin(theta))*e_r^e_theta^e_phi/r'
return
def test_basic_multivector_operations(self):
g3d = Ga('e*x|y|z')
(ex,ey,ez) = g3d.mv()
A = g3d.mv('A','mv')
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
X = g3d.mv('X','vector')
Y = g3d.mv('Y','vector')
assert str(X) == 'X__x*e_x + X__y*e_y + X__z*e_z'
assert str(Y) == 'Y__x*e_x + Y__y*e_y + Y__z*e_z'
assert str((X*Y)) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z + (X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z'
assert str((X^Y)) == '(X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z'
assert str((X|Y)) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z'
g2d = Ga('e*x|y')
(ex,ey) = g2d.mv()
X = g2d.mv('X','vector')
A = g2d.mv('A','spinor')
assert str(X) == 'X__x*e_x + X__y*e_y'
assert str(A) == 'A + A__xy*e_x^e_y'
assert str((X|A)) == '-A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y)*e_x + A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y)*e_y'
assert str((X<A)) == '-A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y)*e_x + A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y)*e_y'
assert str((A>X)) == 'A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y)*e_x - A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y)*e_y'
o2d = Ga('e*x|y', g=[1, 1])
(ex, ey) = o2d.mv()
X = o2d.mv('X', 'vector')
A = o2d.mv('A', 'spinor')
assert str(X) == 'X__x*e_x + X__y*e_y'
assert str(A) == 'A + A__xy*e_x^e_y'
assert str((X*A)) == '(A*X__x - A__xy*X__y)*e_x + (A*X__y + A__xy*X__x)*e_y'
assert str((X|A)) == '-A__xy*X__y*e_x + A__xy*X__x*e_y'
assert str((X<A)) == '-A__xy*X__y*e_x + A__xy*X__x*e_y'
assert str((X>A)) == 'A*X__x*e_x + A*X__y*e_y'
assert str((A*X)) == '(A*X__x + A__xy*X__y)*e_x + (A*X__y - A__xy*X__x)*e_y'
assert str((A|X)) == 'A__xy*X__y*e_x - A__xy*X__x*e_y'
assert str((A<X)) == 'A*X__x*e_x + A*X__y*e_y'
assert str((A>X)) == 'A__xy*X__y*e_x - A__xy*X__x*e_y'
return
def test_basic_multivector_operations(self):
g3d = Ga('e*x|y|z')
(ex,ey,ez) = g3d.mv()
A = g3d.mv('A','mv')
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
assert str(A) == 'A + A__x*e_x + A__y*e_y + A__z*e_z + A__xy*e_x^e_y + A__xz*e_x^e_z + A__yz*e_y^e_z + A__xyz*e_x^e_y^e_z'
X = g3d.mv('X','vector')
Y = g3d.mv('Y','vector')
assert str(X) == 'X__x*e_x + X__y*e_y + X__z*e_z'
assert str(Y) == 'Y__x*e_x + Y__y*e_y + Y__z*e_z'
assert str((X*Y)) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z + (X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z'
assert str((X^Y)) == '(X__x*Y__y - X__y*Y__x)*e_x^e_y + (X__x*Y__z - X__z*Y__x)*e_x^e_z + (X__y*Y__z - X__z*Y__y)*e_y^e_z'
assert str((X|Y)) == '(e_x.e_x)*X__x*Y__x + (e_x.e_y)*X__x*Y__y + (e_x.e_y)*X__y*Y__x + (e_x.e_z)*X__x*Y__z + (e_x.e_z)*X__z*Y__x + (e_y.e_y)*X__y*Y__y + (e_y.e_z)*X__y*Y__z + (e_y.e_z)*X__z*Y__y + (e_z.e_z)*X__z*Y__z'
g2d = Ga('e*x|y')
(ex,ey) = g2d.mv()
X = g2d.mv('X','vector')
A = g2d.mv('A','spinor')
assert str(X) == 'X__x*e_x + X__y*e_y'
assert str(A) == 'A + A__xy*e_x^e_y'
assert str((X|A)) == '-A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y)*e_x + A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y)*e_y'
assert str((X<A)) == '-A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y)*e_x + A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y)*e_y'
assert str((A>X)) == 'A__xy*((e_x.e_y)*X__x + (e_y.e_y)*X__y)*e_x - A__xy*((e_x.e_x)*X__x + (e_x.e_y)*X__y)*e_y'
o2d = Ga('e*x|y', g=[1, 1])
(ex, ey) = o2d.mv()
X = o2d.mv('X', 'vector')
A = o2d.mv('A', 'spinor')
assert str(X) == 'X__x*e_x + X__y*e_y'
assert str(A) == 'A + A__xy*e_x^e_y'
assert str((X*A)) == '(A*X__x - A__xy*X__y)*e_x + (A*X__y + A__xy*X__x)*e_y'
assert str((X|A)) == '-A__xy*X__y*e_x + A__xy*X__x*e_y'
assert str((X<A)) == '-A__xy*X__y*e_x + A__xy*X__x*e_y'
assert str((X>A)) == 'A*X__x*e_x + A*X__y*e_y'
assert str((A*X)) == '(A*X__x + A__xy*X__y)*e_x + (A*X__y - A__xy*X__x)*e_y'
assert str((A|X)) == 'A__xy*X__y*e_x - A__xy*X__x*e_y'
assert str((A<X)) == 'A*X__x*e_x + A*X__y*e_y'
assert str((A>X)) == 'A__xy*X__y*e_x - A__xy*X__x*e_y'
return
def derivatives_in_spherical_coordinates():
Print_Function()
X = (r, th, phi) = symbols('r theta phi')
s3d = Ga('e_r e_theta e_phi', g=[1, r ** 2, r ** 2 * sin(th) ** 2], coords=X, norm=True)
(er, eth, ephi) = s3d.mv()
grad = s3d.grad
f = s3d.mv('f', 'scalar', f=True)
A = s3d.mv('A', 'vector', f=True)
B = s3d.mv('B', 'bivector', f=True)
print('f =', f)
print('A =', A)
print('B =', B)
print('grad*f =', grad * f)
print('grad|A =', grad | A)
print('-I*(grad^A) =', -s3d.E() * (grad ^ A))
print('grad^B =', grad ^ B)
return
def rounding_numerical_components():
Print_Function()
o3d = Ga('e_x e_y e_z', g=[1, 1, 1])
(ex, ey, ez) = o3d.mv()
X = 1.2 * ex + 2.34 * ey + 0.555 * ez
Y = 0.333 * ex + 4 * ey + 5.3 * ez
print('X =', X)
print('Nga(X,2) =', Nga(X, 2))
print('X*Y =', X * Y)
print('Nga(X*Y,2) =', Nga(X * Y, 2))
return
def basic_multivector_operations_2D_orthogonal():
Print_Function()
o2d = Ga('e*x|y', g=[1, 1])
(ex, ey) = o2d.mv()
print('g_{ii} =', o2d.g)
X = o2d.mv('X', 'vector')
A = o2d.mv('A', 'spinor')
print(X.Fmt(1, 'X'))
print(A.Fmt(1, 'A'))
print((X * A).Fmt(2, 'X*A'))
print((X | A).Fmt(2, 'X|A'))
print((X < A).Fmt(2, 'X<A'))
print((X > A).Fmt(2, 'X>A'))
print((A * X).Fmt(2, 'A*X'))
print((A | X).Fmt(2, 'A|X'))
print((A < X).Fmt(2, 'A<X'))
print((A > X).Fmt(2, 'A>X'))
return
def test_rounding_numerical_components(self):
o3d = Ga('e_x e_y e_z', g=[1, 1, 1])
(ex, ey, ez) = o3d.mv()
X = 1.2*ex+2.34*ey+0.555*ez
Y = 0.333*ex+4*ey+5.3*ez
assert str(X) == '1.2*e_x + 2.34*e_y + 0.555*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 + 4.02078*e_x^e_y + 6.175185*e_x^e_z + 10.182*e_y^e_z'
assert str(Nga(X*Y,2)) == '13. + 4.0*e_x^e_y + 6.2*e_x^e_z + 10.0*e_y^e_z'
return
def test_lt_function(self):
""" Test construction from a function """
base = Ga('a b', g=[1, 1], coords=symbols('x, y', real=True))
a, b = base.mv()
def not_linear(x):
return x * x
with pytest.raises(ValueError, match='linear'):
base.lt(not_linear)
def not_vector(x):
return x + S.One
with pytest.raises(ValueError, match='vector'):
base.lt(not_vector)
def ok(x):
return (x | b) * a + 2*x
f = base.lt(ok)
x = base.mv('x', 'vector')
y = base.mv('y', 'vector')
assert f(x) == ok(x)
assert f(x^y) == ok(x)^ok(y)
assert f(1 + 2*(x^y)) == 1 + 2*(ok(x)^ok(y))
def test_rounding_numerical_components(self):
o3d = Ga('e_x e_y e_z', g=[1, 1, 1])
(ex, ey, ez) = o3d.mv()
X = 1.2*ex+2.34*ey+0.555*ez
Y = 0.333*ex+4*ey+5.3*ez
assert str(X) == '1.2*e_x + 2.34*e_y + 0.555*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 + 4.02078*e_x^e_y + 6.175185*e_x^e_z + 10.182*e_y^e_z'
assert str(Nga(X*Y,2)) == '13. + 4.0*e_x^e_y + 6.2*e_x^e_z + 10.0*e_y^e_z'
return
def conformal_representations_of_circles_lines_spheres_and_planes():
global n, nbar
Print_Function()
g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0'
cnfml3d = Ga('e_1 e_2 e_3 n nbar', g=g)
(e1, e2, e3, n, nbar) = cnfml3d.mv()
print('g_{ij} =\n', cnfml3d.g)
e = n + nbar
#conformal representation of points
A = make_vector(e1, ga=cnfml3d) # point a = (1,0,0) A = F(a)
B = make_vector(e2, ga=cnfml3d) # point b = (0,1,0) B = F(b)
C = make_vector(-e1, ga=cnfml3d) # point c = (-1,0,0) C = F(c)
D = make_vector(e3, ga=cnfml3d) # point d = (0,0,1) D = F(d)
X = make_vector('x', 3, ga=cnfml3d)
print('F(a) =', A)
print('F(b) =', B)
print('F(c) =', C)
print('F(d) =', D)
print('F(x) =', X)
print('a = e1, b = e2, c = -e1, and d = e3')
print('A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.')
print('Circle through a, b, and c')
print('Circle: A^B^C^X = 0 =', (A ^ B ^ C ^ X))
print('Line through a and b')
print('Line : A^B^n^X = 0 =', (A ^ B ^ n ^ X))
print('Sphere through a, b, c, and d')
print('Sphere: A^B^C^D^X = 0 =', (((A ^ B) ^ C) ^ D) ^ X)
print('Plane through a, b, and d')
print('Plane : A^B^n^D^X = 0 =', (A ^ B ^ n ^ D ^ X))
L = (A ^ B ^ e) ^ X
print(L.Fmt(3, 'Hyperbolic Circle: (A^B^e)^X = 0 ='))
return
def check_generalized_BAC_CAB_formulas():
Print_Function()
g5d = Ga('a b c d e')
(a, b, c, d, e) = g5d.mv()
print('g_{ij} =\n', g5d.g)
print('a|(b*c) =', a | (b * c))
print('a|(b^c) =', a | (b ^ c))
print('a|(b^c^d) =', a | (b ^ c ^ d))
print('a|(b^c)+c|(a^b)+b|(c^a) =', (a | ( b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)))
print('a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b))
print('a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c) =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c))
print('(a^b)|(c^d) =',(a^b)|(c^d))
print('((a^b)|c)|d =',((a^b)|c)|d)
print('(a^b)x(c^d) =',Ga.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()
g4d = Ga('a b c d e')
(a, b, c, d, e) = g4d.mv()
print('g_{ij} =', g4d.g)
print('\\bm{a|(b*c)} =', a | (b * c))
print('\\bm{a|(b^c)} =', a | (b ^ c))
print('\\bm{a|(b^c^d)} =', a | (b ^ c ^ d))
print('\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',
(a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)))
print('\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',
a * (b ^ c) - b * (a ^ c) + c * (a ^ b))
print(
'\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',
a * (b ^ c ^ d) - b * (a ^ c ^ d) + c * (a ^ b ^ d) - d * (a ^ b ^ c))
print('\\bm{(a^b)|(c^d)} =', (a ^ b) | (c ^ d))
print('\\bm{((a^b)|c)|d} =', ((a ^ b) | c) | d)
print('\\bm{(a^b)\\times (c^d)} =', Ga.com(a ^ b, c ^ d))
print('\\bm{(a^b^c)(d^e)} =', ((a ^ b ^ c) * (d ^ e)).Fmt(2))
return
def main():
Format()
snr = 1
g = '0 0 1 0 ,0 0 0 1 ,1 0 0 0 ,0 1 0 0'
sk4coords = (e1, e2, e3, e4) = symbols('e1 e2 e3 e4')
sk4 = Ga('e_1 e_2 e_3 e_4', g=g, coords=sk4coords)
(e1, e2, e3, e4) = sk4.mv()
print('g_{ii} =', sk4.g)
v = symbols('v', real=True)
x1 = (e1 + e3) / sqrt(2)
x2 = (e2 + e4) / sqrt(2)
print('x_1<x_1==', x1 < x1)
print('x_1<x_2==', x1 < x2)
print('x_2<x_1==', x2 < x1)
print('x_2<x_2==', x2 < x2)
print(r'#$-\infty < v < \infty$')
print('(-v*(x_1^x_2)/2).exp()==', (-v * (x1 ^ x2) / 2).exp())
v = symbols('v', real=True, positive=True)
print(r'#$0\le v < \infty$')
print('(-v*(x_1^x_2)/2).exp()==', (-v * (x1 ^ x2) / 2).exp())
xpdf(pdfprog=None)
return
def test_reciprocal_frame_test(self):
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
g3dn = Ga('e1 e2 e3',g=g)
(e1,e2,e3) = g3dn.mv()
E = e1^e2^e3
Esq = (E*E).scalar()
assert str(E) == 'e1^e2^e3'
assert str(Esq) == '(e1.e2)**2 - 2*(e1.e2)*(e1.e3)*(e2.e3) + (e1.e3)**2 + (e2.e3)**2 - 1'
Esq_inv = 1/Esq
E1 = (e2^e3)*E
E2 = (-1)*(e1^e3)*E
E3 = (e1^e2)*E
assert str(E1) == '((e2.e3)**2 - 1)*e1 + ((e1.e2) - (e1.e3)*(e2.e3))*e2 + (-(e1.e2)*(e2.e3) + (e1.e3))*e3'
assert str(E2) == '((e1.e2) - (e1.e3)*(e2.e3))*e1 + ((e1.e3)**2 - 1)*e2 + (-(e1.e2)*(e1.e3) + (e2.e3))*e3'
assert str(E3) == '(-(e1.e2)*(e2.e3) + (e1.e3))*e1 + (-(e1.e2)*(e1.e3) + (e2.e3))*e2 + ((e1.e2)**2 - 1)*e3'
w = (E1|e2)
w = w.expand()
assert str(w) == '0'
w = (E1|e3)
w = w.expand()
assert str(w) == '0'
w = (E2|e1)
w = w.expand()
assert str(w) == '0'
w = (E2|e3)
w = w.expand()
assert str(w) == '0'
w = (E3|e1)
w = w.expand()
assert str(w) == '0'
w = (E3|e2)
w = w.expand()
assert str(w) == '0'
w = (E1|e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
assert str(simplify(w/Esq)) == '1'
w = (E2|e2)
w = (w.expand()).scalar()
assert str(simplify(w/Esq)) == '1'
w = (E3|e3)
w = (w.expand()).scalar()
assert str(simplify(w/Esq)) == '1'
return
def main():
# Print_Function()
(x, y, z) = xyz = symbols('x,y,z',real=True)
(o3d, ex, ey, ez) = Ga.build('e_x e_y e_z', g=[1, 1, 1], coords=xyz)
grad = o3d.grad
(u, v) = uv = symbols('u,v',real=True)
(g2d, eu, ev) = Ga.build('e_u e_v', coords=uv)
grad_uv = g2d.grad
v_xyz = o3d.mv('v','vector')
A_xyz = o3d.mv('A','vector',f=True)
A_uv = g2d.mv('A','vector',f=True)
print('#3d orthogonal ($A$ is vector function)')
print('A =', A_xyz)
print('%A^{2} =', A_xyz * A_xyz)
print('grad|A =', grad | A_xyz)
print('grad*A =', grad * A_xyz)
print('v|(grad*A) =',v_xyz|(grad*A_xyz))
print('#2d general ($A$ is vector function)')
print('A =', A_uv)
print('%A^{2} =', A_uv * A_uv)
print('grad|A =', grad_uv | A_uv)
print('grad*A =', grad_uv * A_uv)
A = o3d.lt('A')
print('#3d orthogonal ($A,\\;B$ are linear transformations)')
print('A =', A)
print(r'\f{mat}{A} =', A.matrix())
print('\\f{\\det}{A} =', A.det())
print('\\overline{A} =', A.adj())
print('\\f{\\Tr}{A} =', A.tr())
print('\\f{A}{e_x^e_y} =', A(ex^ey))
print('\\f{A}{e_x}^\\f{A}{e_y} =', A(ex)^A(ey))
B = o3d.lt('B')
print('g =', o3d.g)
print('%g^{-1} =', latex(o3d.g_inv))
print('A + B =', A + B)
print('AB =', A * B)
print('A - B =', A - B)
print('General Symmetric Linear Transformation')
Asym = o3d.lt('A',mode='s')
print('A =', Asym)
print('General Antisymmetric Linear Transformation')
Aasym = o3d.lt('A',mode='a')
print('A =', Aasym)
print('#2d general ($A,\\;B$ are linear transformations)')
A2d = g2d.lt('A')
print('g =', g2d.g)
print('%g^{-1} =', latex(g2d.g_inv))
print('%gg^{-1} =', latex(simplify(g2d.g * g2d.g_inv)))
print('A =', A2d)
print(r'\f{mat}{A} =', A2d.matrix())
print('\\f{\\det}{A} =', A2d.det())
A2d_adj = A2d.adj()
print('\\overline{A} =', A2d_adj)
print('\\f{mat}{\\overline{A}} =', latex(simplify(A2d_adj.matrix())))
print('\\f{\\Tr}{A} =', A2d.tr())
print('\\f{A}{e_u^e_v} =', A2d(eu^ev))
print('\\f{A}{e_u}^\\f{A}{e_v} =', A2d(eu)^A2d(ev))
B2d = g2d.lt('B')
print('B =', B2d)
print('A + B =', A2d + B2d)
print('A - B =', A2d - B2d)
# TODO: add this back when we drop Sympy 1.3. The 64kB of output is far too
# printer-dependent
if False:
print('AB =', A2d * B2d)
a = g2d.mv('a','vector')
b = g2d.mv('b','vector')
print(r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',((a|A2d.adj()(b))-(b|A2d(a))).simplify())
m4d = Ga('e_t e_x e_y e_z', g=[1, -1, -1, -1],coords=symbols('t,x,y,z',real=True))
T = m4d.lt('T')
print('g =', m4d.g)
print(r'\underline{T} =',T)
print(r'\overline{T} =',T.adj())
print(r'\f{\det}{\underline{T}} =',T.det())
print(r'\f{\mbox{tr}}{\underline{T}} =',T.tr())
a = m4d.mv('a','vector')
b = m4d.mv('b','vector')
print(r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',((a|T.adj()(b))-(b|T(a))).simplify())
coords = (r, th, phi) = symbols('r,theta,phi', real=True)
(sp3d, er, eth, ephi) = Ga.build('e_r e_th e_ph', g=[1, r**2, r**2*sin(th)**2], coords=coords)
grad = sp3d.grad
sm_coords = (u, v) = symbols('u,v', real=True)
smap = [1, u, v] # Coordinate map for sphere of r = 1
sph2d = sp3d.sm(smap,sm_coords,norm=True)
(eu, ev) = sph2d.mv()
grad_uv = sph2d.grad
F = sph2d.mv('F','vector',f=True)
f = sph2d.mv('f','scalar',f=True)
print('f =',f)
print('grad*f =',grad_uv * f)
print('F =',F)
print('grad*F =',grad_uv * F)
tp = (th,phi) = symbols('theta,phi',real=True)
smap = [sin(th)*cos(phi),sin(th)*sin(phi),cos(th)]
sph2dr = o3d.sm(smap,tp,norm=True)
(eth, ephi) = sph2dr.mv()
grad_tp = sph2dr.grad
F = sph2dr.mv('F','vector',f=True)
f = sph2dr.mv('f','scalar',f=True)
print('f =',f)
print('grad*f =',grad_tp * f)
print('F =',F)
print('grad*F =',grad_tp * F)
return
def test_lt_matrix(self):
base = Ga('a b', g=[1, 1], coords=symbols('x, y', real=True))
a, b = base.mv()
A = base.lt([a+b, 2*a-b])
assert str(A) == 'Lt(a) = a + b\nLt(b) = 2*a - b'
assert str(A.matrix()) == 'Matrix([[1, 2], [1, -1]])'
from __future__ import absolute_import, division
from __future__ import print_function
from sympy import expand, simplify
from galgebra.printer import Format, xpdf
from galgebra.ga import Ga
g = '1 # #,' + \
'# 1 #,' + \
'# # 1'
Format()
ng3d = Ga('e1 e2 e3', g=g)
(e1, e2, e3) = ng3d.mv()
print('g_{ij} =', ng3d.g)
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()
# A2d.adj().Fmt(4,'\\overline{A}')
print('\\f{\\Tr}{A} =', A2d.tr())
print('\\f{A}{e_u^e_v} =', A2d(eu ^ ev))
print('\\f{A}{e_u}^\\f{A}{e_v} =', A2d(eu) ^ A2d(ev))
B2d = g2d.lt('B')
print('B =', B2d)
print('A + B =', A2d + B2d)
print('AB =', A2d * B2d)
print('A - B =', A2d - B2d)
a = g2d.mv('a', 'vector')
b = g2d.mv('b', 'vector')
print(r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',
((a | A2d.adj()(b)) - (b | A2d(a))).simplify())
print('#4d Minkowski spaqce (Space Time)')
m4d = Ga('e_t e_x e_y e_z',
g=[1, -1, -1, -1],
coords=symbols('t,x,y,z', real=True))
T = m4d.lt('T')
print('g =', m4d.g)
# FIXME incorrect sign for T and T.adj()
print(r'\underline{T} =', T)
print(r'\overline{T} =', T.adj())
# m4d.mv(T.det()).Fmt(4,r'\f{\det}{\underline{T}}')
print(r'\f{\mbox{tr}}{\underline{T}} =', T.tr())
a = m4d.mv('a', 'vector')
b = m4d.mv('b', 'vector')
print(r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',
((a | T.adj()(b)) - (b | T(a))).simplify())
xpdf(paper=(10, 12), debug=True)
def basic_multivector_operations():
Print_Function()
g3d = Ga('e*x|y|z')
(ex, ey, ez) = g3d.mv()
A = g3d.mv('A', 'mv')
print(A.Fmt(1, 'A'))
print(A.Fmt(2, 'A'))
print(A.Fmt(3, 'A'))
X = g3d.mv('X', 'vector')
Y = g3d.mv('Y', 'vector')
print('g_{ij} =\n', g3d.g)
print(X.Fmt(1, 'X'))
print(Y.Fmt(1, 'Y'))
print((X * Y).Fmt(2, 'X*Y'))
print((X ^ Y).Fmt(2, 'X^Y'))
print((X | Y).Fmt(2, 'X|Y'))
g2d = Ga('e*x|y')
(ex, ey) = g2d.mv()
print('g_{ij} =\n', g2d.g)
X = g2d.mv('X', 'vector')
A = g2d.mv('A', 'spinor')
print(X.Fmt(1, 'X'))
print(A.Fmt(1, 'A'))
print((X | A).Fmt(2, 'X|A'))
print((X < A).Fmt(2, 'X<A'))
print((A > X).Fmt(2, 'A>X'))
o2d = Ga('e*x|y', g=[1, 1])
(ex, ey) = o2d.mv()
print('g_{ii} =\n', o2d.g)
X = o2d.mv('X', 'vector')
A = o2d.mv('A', 'spinor')
print(X.Fmt(1, 'X'))
print(A.Fmt(1, 'A'))
print((X * A).Fmt(2, 'X*A'))
print((X | A).Fmt(2, 'X|A'))
print((X < A).Fmt(2, 'X<A'))
print((X > A).Fmt(2, 'X>A'))
print((A * X).Fmt(2, 'A*X'))
print((A | X).Fmt(2, 'A|X'))
print((A < X).Fmt(2, 'A<X'))
print((A > X).Fmt(2, 'A>X'))
return
def reciprocal_frame_test():
Print_Function()
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
g3dn = Ga('e1 e2 e3', g=g)
(e1, e2, e3) = g3dn.mv()
print('g_{ij} =\n', g3dn.g)
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():
from sympy import solve, sqrt
Print_Function()
g = '0 # #,# 0 #,# # 1'
necl = Ga('X Y e', g=g)
(X, Y, e) = necl.mv()
print('g_{ij} =', necl.g)
print('(X^Y)**2 =', (X ^ Y) * (X ^ Y))
L = X ^ Y ^ e
B = (L * e).expand().blade_rep() # D&L 10.152
print('B =', B)
Bsq = B * B
print('B**2 =', Bsq.obj)
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) = symbols('s c (1/B) M S C alpha')
XdotY = necl.g[0, 1]
Xdote = necl.g[0, 2]
Ydote = necl.g[1, 2]
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])
print(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(Binv, 1 / 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 test_noneuclidian_distance_calculation(self):
from sympy import solve, sqrt
g = '0 # #,# 0 #,# # 1'
necl = Ga('X Y e', g=g)
(X, Y, e) = necl.mv()
assert str((X ^ Y) * (X ^ Y)) == '(X.Y)**2'
L = X ^ Y ^ e
B = (L * e).expand().blade_rep() # D&L 10.152
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
Bsq = B * B
assert str(Bsq) == '(X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e)'
Bsq = Bsq.scalar()
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
BeBr = B * e * B.rev()
assert str(BeBr) == '(X.Y)*(-(X.Y) + 2*(X.e)*(Y.e))*e'
assert str(B * B) == '(X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e)'
assert str(L * L) == '(X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e)' # D&L 10.153
(s, c, Binv, M, S, C, alpha) = symbols('s c (1/B) M S C alpha')
XdotY = necl.g[0, 1]
Xdote = necl.g[0, 2]
Ydote = necl.g[1, 2]
Bhat = Binv * B # D&L 10.154
R = c + s * Bhat # Rotor R = exp(alpha*Bhat/2)
assert str(
R) == 'c + (1/B)*s*X^Y - (1/B)*(Y.e)*s*X^e + (1/B)*(X.e)*s*Y^e'
Z = R * X * R.rev() # D&L 10.155
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv, s, c, XdotY])
assert str(
Z
) == '((1/B)**2*(X.Y)**2*s**2 - 2*(1/B)**2*(X.Y)*(X.e)*(Y.e)*s**2 + 2*(1/B)*(X.Y)*c*s - 2*(1/B)*(X.e)*(Y.e)*c*s + c**2)*X + 2*(1/B)*(X.e)**2*c*s*Y + 2*(1/B)*(X.Y)*(X.e)*s*(-(1/B)*(X.Y)*s + 2*(1/B)*(X.e)*(Y.e)*s - c)*e'
W = Z | Y
# From this point forward all calculations are with sympy scalars
W = W.scalar()
assert str(
W
) == '(1/B)**2*(X.Y)**3*s**2 - 4*(1/B)**2*(X.Y)**2*(X.e)*(Y.e)*s**2 + 4*(1/B)**2*(X.Y)*(X.e)**2*(Y.e)**2*s**2 + 2*(1/B)*(X.Y)**2*c*s - 4*(1/B)*(X.Y)*(X.e)*(Y.e)*c*s + (X.Y)*c**2'
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(Binv, 1 / Bmag)
W = expand(W)
assert str(
W.simplify()
) == '(X.Y)*C - (X.e)*(Y.e)*C + (X.e)*(Y.e) + S*sqrt((X.Y)*((X.Y) - 2*(X.e)*(Y.e)))'
Wd = collect(W, [C, S], exact=True, evaluate=False)
Wd_1 = Wd[one]
Wd_C = Wd[C]
Wd_S = Wd[S]
assert str(Wd_1) == '(X.e)*(Y.e)'
assert str(Wd_C) == '(X.Y) - (X.e)*(Y.e)'
assert str(Wd_S) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
assert str(Bmag) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
Wd_1 = Wd_1.subs(Binv, 1 / Bmag)
Wd_C = Wd_C.subs(Binv, 1 / Bmag)
Wd_S = Wd_S.subs(Binv, 1 / Bmag)
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])
assert str(a) == '(X.e)**2*(Y.e)**2'
assert str(b) == '2*(X.e)*(Y.e)*((X.Y) - (X.e)*(Y.e))'
assert str(c) == '(X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e) + (X.e)**2*(Y.e)**2'
x = Symbol('x')
C = solve(a * x**2 + b * x + c, x)[0]
assert str(expand(simplify(expand(C)))) == '-(X.Y)/((X.e)*(Y.e)) + 1'
def test_noneuclidian_distance_calculation(self):
from sympy import solve,sqrt
g = '0 # #,# 0 #,# # 1'
necl = Ga('X Y e',g=g)
(X,Y,e) = necl.mv()
assert str((X^Y)*(X^Y)) == '(X.Y)**2'
L = X^Y^e
B = (L*e).expand().blade_rep() # D&L 10.152
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
Bsq = B*B
assert str(Bsq) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
Bsq = Bsq.scalar()
assert str(B) == 'X^Y - (Y.e)*X^e + (X.e)*Y^e'
BeBr = B*e*B.rev()
assert str(BeBr) == '(X.Y)*(-(X.Y) + 2*(X.e)*(Y.e))*e'
assert str(B*B) == '(X.Y)*((X.Y) - 2*(X.e)*(Y.e))'
assert str(L*L) == '(X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e)' # D&L 10.153
(s,c,Binv,M,S,C,alpha) = symbols('s c (1/B) M S C alpha')
XdotY = necl.g[0,1]
Xdote = necl.g[0,2]
Ydote = necl.g[1,2]
Bhat = Binv*B # D&L 10.154
R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
assert str(R) == 'c + (1/B)*s*X^Y - (1/B)*(Y.e)*s*X^e + (1/B)*(X.e)*s*Y^e'
Z = R*X*R.rev() # D&L 10.155
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv,s,c,XdotY])
assert str(Z) == '((1/B)**2*(X.Y)**2*s**2 - 2*(1/B)**2*(X.Y)*(X.e)*(Y.e)*s**2 + 2*(1/B)*(X.Y)*c*s - 2*(1/B)*(X.e)*(Y.e)*c*s + c**2)*X + 2*(1/B)*(X.e)**2*c*s*Y + 2*(1/B)*(X.Y)*(X.e)*s*(-(1/B)*(X.Y)*s + 2*(1/B)*(X.e)*(Y.e)*s - c)*e'
W = Z|Y
# From this point forward all calculations are with sympy scalars
W = W.scalar()
assert str(W) == '(1/B)**2*(X.Y)**3*s**2 - 4*(1/B)**2*(X.Y)**2*(X.e)*(Y.e)*s**2 + 4*(1/B)**2*(X.Y)*(X.e)**2*(Y.e)**2*s**2 + 2*(1/B)*(X.Y)**2*c*s - 4*(1/B)*(X.Y)*(X.e)*(Y.e)*c*s + (X.Y)*c**2'
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(Binv,1/Bmag)
W = expand(W)
assert str(W.simplify()) == '(X.Y)*C - (X.e)*(Y.e)*C + (X.e)*(Y.e) + S*sqrt((X.Y)*((X.Y) - 2*(X.e)*(Y.e)))'
Wd = collect(W,[C,S],exact=True,evaluate=False)
Wd_1 = Wd[one]
Wd_C = Wd[C]
Wd_S = Wd[S]
assert str(Wd_1) == '(X.e)*(Y.e)'
assert str(Wd_C) == '(X.Y) - (X.e)*(Y.e)'
assert str(Wd_S) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
assert str(Bmag) == 'sqrt((X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e))'
Wd_1 = Wd_1.subs(Binv,1/Bmag)
Wd_C = Wd_C.subs(Binv,1/Bmag)
Wd_S = Wd_S.subs(Binv,1/Bmag)
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])
assert str(a) == '(X.e)**2*(Y.e)**2'
assert str(b) == '2*(X.e)*(Y.e)*((X.Y) - (X.e)*(Y.e))'
assert str(c) == '(X.Y)**2 - 2*(X.Y)*(X.e)*(Y.e) + (X.e)**2*(Y.e)**2'
x = Symbol('x')
C = solve(a*x**2+b*x+c,x)[0]
assert str(expand(simplify(expand(C)))) == '-(X.Y)/((X.e)*(Y.e)) + 1'
return
def main():
Format()
(g3d, ex, ey, ez) = Ga.build('e*x|y|z')
A = g3d.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', real=True)
o3d = Ga('e_x e_y e_z', g=[1, 1, 1], coords=X)
(ex, ey, ez) = o3d.mv()
f = o3d.mv('f', 'scalar', f=True)
A = o3d.mv('A', 'vector', f=True)
B = o3d.mv('B', 'bivector', f=True)
print(r'\bm{A} =', A)
print(r'\bm{B} =', B)
print('grad*f =', o3d.grad * f)
print(r'grad|\bm{A} =', o3d.grad | A)
print(r'grad*\bm{A} =', o3d.grad * A)
print(r'-I*(grad^\bm{A}) =', -o3d.i * (o3d.grad ^ A))
print(r'grad*\bm{B} =', o3d.grad * B)
print(r'grad^\bm{B} =', o3d.grad ^ B)
print(r'grad|\bm{B} =', o3d.grad | B)
g4d = Ga('a b c d')
(a, b, c, d) = g4d.mv()
print('g_{ij} =', g4d.g)
print('\\bm{a|(b*c)} =', a | (b * c))
print('\\bm{a|(b^c)} =', a | (b ^ c))
print('\\bm{a|(b^c^d)} =', a | (b ^ c ^ d))
print('\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',
(a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)))
print('\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',
a * (b ^ c) - b * (a ^ c) + c * (a ^ b))
print(
'\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',
a * (b ^ c ^ d) - b * (a ^ c ^ d) + c * (a ^ b ^ d) - d * (a ^ b ^ c))
print('\\bm{(a^b)|(c^d)} =', (a ^ b) | (c ^ d))
print('\\bm{((a^b)|c)|d} =', ((a ^ b) | c) | d)
print('\\bm{(a^b)\\times (c^d)} =', Ga.com(a ^ b, c ^ d))
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
ng3d = Ga('e1 e2 e3', g=g)
(e1, e2, e3) = ng3d.mv()
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')
s3d = Ga('e_r e_theta e_phi',
g=[1, r**2, r**2 * sin(th)**2],
coords=X,
norm=True)
(er, eth, ephi) = s3d.mv()
f = s3d.mv('f', 'scalar', f=True)
A = s3d.mv('A', 'vector', f=True)
B = s3d.mv('B', 'bivector', f=True)
print('A =', A)
print('B =', B)
print('grad*f =', s3d.grad * f)
print('grad|A =', s3d.grad | A)
print('-I*(grad^A) =', -s3d.i * (s3d.grad ^ A))
print('grad^B =', s3d.grad ^ B)
coords = symbols('t x y z')
m4d = Ga('gamma*t|x|y|z', g=[1, -1, -1, -1], coords=coords)
(g0, g1, g2, g3) = m4d.mv()
I = m4d.i
B = m4d.mv('B', 'vector', f=True)
E = m4d.mv('E', 'vector', f=True)
B.set_coef(1, 0, 0)
E.set_coef(1, 0, 0)
B *= g0
E *= g0
J = m4d.mv('J', 'vector', f=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 = m4d.grad * F
gradF.Fmt(3, 'grad*F')
print('grad*F = J')
(gradF.get_grade(1) - J).Fmt(3, '%\\grade{\\nabla F}_{1} -J = 0')
(gradF.get_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'")
m2d = Ga('gamma*t|x', g=[1, -1])
(g0, g1) = m2d.mv()
R = cosh(alpha / 2) + sinh(alpha / 2) * (g0 ^ g1)
X = t * g0 + x * g1
Xp = tp * g0 + xp * g1
print('R =', R)
print(
r"#%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} = t'\bm{\gamma'_{t}}+x'\bm{\gamma'_{x}} = R\lp t'\bm{\gamma_{t}}+x'\bm{\gamma_{x}}\rp R^{\dagger}"
)
Xpp = R * Xp * R.rev()
Xpp = Xpp.collect()
Xpp = Xpp.trigsimp()
print(r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =", Xpp)
Xpp = Xpp.subs({sinh(alpha): gamma * beta, cosh(alpha): gamma})
print(r'%\f{\sinh}{\alpha} = \gamma\beta')
print(r'%\f{\cosh}{\alpha} = \gamma')
print(r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =", Xpp.collect())
coords = symbols('t x y z')
m4d = Ga('gamma*t|x|y|z', g=[1, -1, -1, -1], coords=coords)
(g0, g1, g2, g3) = m4d.mv()
I = m4d.i
(m, e) = symbols('m e')
psi = m4d.mv('psi', 'spinor', f=True)
A = m4d.mv('A', 'vector', f=True)
sig_z = g3 * g0
print('\\bm{A} =', A)
print('\\bm{\\psi} =', psi)
dirac_eq = (m4d.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
def test_reciprocal_frame_test(self):
g = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
g3dn = Ga('e1 e2 e3', g=g)
(e1, e2, e3) = g3dn.mv()
E = e1 ^ e2 ^ e3
Esq = (E * E).scalar()
assert str(E) == 'e1^e2^e3'
assert str(
Esq
) == '(e1.e2)**2 - 2*(e1.e2)*(e1.e3)*(e2.e3) + (e1.e3)**2 + (e2.e3)**2 - 1'
Esq_inv = 1 / Esq
E1 = (e2 ^ e3) * E
E2 = (-1) * (e1 ^ e3) * E
E3 = (e1 ^ e2) * E
assert str(
E1
) == '((e2.e3)**2 - 1)*e1 + ((e1.e2) - (e1.e3)*(e2.e3))*e2 + (-(e1.e2)*(e2.e3) + (e1.e3))*e3'
assert str(
E2
) == '((e1.e2) - (e1.e3)*(e2.e3))*e1 + ((e1.e3)**2 - 1)*e2 + (-(e1.e2)*(e1.e3) + (e2.e3))*e3'
assert str(
E3
) == '(-(e1.e2)*(e2.e3) + (e1.e3))*e1 + (-(e1.e2)*(e1.e3) + (e2.e3))*e2 + ((e1.e2)**2 - 1)*e3'
w = (E1 | e2)
w = w.expand()
assert str(w) == ZERO_STR
w = (E1 | e3)
w = w.expand()
assert str(w) == ZERO_STR
w = (E2 | e1)
w = w.expand()
assert str(w) == ZERO_STR
w = (E2 | e3)
w = w.expand()
assert str(w) == ZERO_STR
w = (E3 | e1)
w = w.expand()
assert str(w) == ZERO_STR
w = (E3 | e2)
w = w.expand()
assert str(w) == ZERO_STR
w = (E1 | e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
assert str(simplify(w / Esq)) == '1'
w = (E2 | e2)
w = (w.expand()).scalar()
assert str(simplify(w / Esq)) == '1'
w = (E3 | e3)
w = (w.expand()).scalar()
assert str(simplify(w / Esq)) == '1'
from __future__ import print_function
import sys
from sympy import symbols, sin, latex, diff, Function, expand
from galgebra.ga import Ga
from galgebra.lt import Mlt
from galgebra.printer import Eprint, Format, xpdf
Format()
#Define spherical coordinate system in 3-d
coords = (r, th, phi) = symbols('r,theta,phi', real=True)
sp3d = Ga('e_r,e_th,e_ph', g=[1, r**2, r**2 * sin(th)**2], coords=coords)
(er, eth, ephi) = sp3d.mv()
#Define coordinates for 2-d (u,v) and 1-d (s) manifolds
u, v, s, alpha = symbols('u v s alpha', real=True)
sub_coords = (u, v)
smap = [1, u, v] # Coordinate map for sphere of r = 1 in 3-d
print(r'(u,v)\rightarrow (r,\theta,\phi) = ', smap)
#Define unit sphere manifold
sph2d = sp3d.sm(smap, sub_coords)
print('#Unit Sphere Manifold:')
