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_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 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
print(r'\bm{\mbox{Two dimensioanal submanifold - Unit sphere}}')
print(r'\text{Basis not normalised}')
sp2coords = (theta, phi) = symbols('theta phi', real = True)
sp2param = [sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)]
sp2 = g3.sm(sp2param, sp2coords, norm = False) # submanifold
(etheta, ephi) = sp2.mv() # sp2 basis vectors
(rtheta, rphi) = sp2.mvr() # sp2 reciprocal basis vectors
sp2grad = sp2.grad
sph_map = [1, theta, phi] # Coordinate map for sphere of r = 1
print(r'(\theta,\phi)\rightarrow (r,\theta,\phi) = ', latex(sph_map))
(etheta, ephi) = sp2.mv()
print(r'e_\theta | e_\theta = ', etheta|etheta)
print(r'e_\phi | e_\phi = ', ephi|ephi)
print('g =',sp2.g)
print(r'\text{g\_inv = }', latex(sp2.g_inv))
#print(r'\text{signature = ', latex(sp2.signature()))
Cf1 = sp2.Christoffel_symbols(mode=1)
Cf1 = permutedims(Array(Cf1), (2, 0, 1))
print(r'\text{Christoffel symbols of the first kind: }')
print(r'\Gamma_{1, \alpha, \beta} = ', latex(Cf1[0, :, :], mat_str=None), r'\quad', r'\Gamma_{2, \alpha, \beta} = ', latex(Cf1[1, :, :], mat_str=None))