f = sph2d.mv('f','vector',f=True)
print 'a =', a
print 'f =', f
print r'%\nabla =', sph2d.grad
#Define directional derivative in direction a for unit sphere manifold
dd = a|sph2d.grad
print r'%a\cdot\nabla =', dd
print r'%\paren{a\cdot\nabla}\bm{e}_u =', dd * eu
print r'%\paren{a\cdot\nabla}\bm{e}_v =', dd * ev
print r'%\paren{a\cdot\nabla}f =',dd * f
V = Mlt('V',sph2d,nargs=1,fct=True)
T = Mlt('T',sph2d,nargs=2,fct=True)
print '#Tensors on the Unit Sphere'
print 'V =', V
#print 'T =', T
T.Fmt(5,'T')
print '#Tensor Contraction'
print r'T[1,2] =', T.contract(1,2)
print '#Tensor Evaluation'
print r'T(a,b) =', T(a,b)
print r'T(a,b+c) =', T(a,b+c).expand()
print r'T(a,\alpha b) =', T(a,alpha*b)
print '#Geometric Derivative With Respect To Slot'
print r'\nabla_{a_{1}}T =',T.pdiff(1)
def test_submanifolds():
#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
#Define unit sphere manifold
sph2d = sp3d.sm(smap,sub_coords)
(eu,ev) = sph2d.mv()
#Define vector and vector field on unit sphere tangent space
a = sph2d.mv('a','vector')
b = sph2d.mv('b','vector')
c = sph2d.mv('c','vector')
f = sph2d.mv('f','vector',f=True)
#Define directional derivative in direction a for unit sphere manifold
dd = a|sph2d.grad
assert str(dd) == 'a__u*D{u} + a__v*D{v}'
assert str(dd * eu) == 'a__v*e_v/tan(u)'
assert str(dd * ev) == '-a__v*sin(2*u)*e_u/2 + a__u*e_v/tan(u)'
assert str(dd * f) == '(a__u*D{u}f__u - a__v*f__v*sin(2*u)/2 + a__v*D{v}f__u)*e_u + (a__u*f__v/tan(u) + a__u*D{u}f__v + a__v*f__u/tan(u) + a__v*D{v}f__v)*e_v'
V = Mlt('V',sph2d,nargs=1,fct=True)
T = Mlt('T',sph2d,nargs=2,fct=True)
assert str(T.contract(1,2)) == 'a_1__u**2*D{u}^2T_uu + a_1__u**2*D{v}^2T_uu/sin(u)**2 + a_1__u*a_1__v*D{u}^2T_uv + a_1__u*a_1__v*D{u}^2T_vu + a_1__u*a_1__v*D{v}^2T_uv/sin(u)**2 + a_1__u*a_1__v*D{v}^2T_vu/sin(u)**2 + a_1__v**2*D{u}^2T_vv + a_1__v**2*D{v}^2T_vv/sin(u)**2'
#Tensor Evaluation
assert str(T(a,b)) == 'a__u*b__u*T_uu + a__u*b__v*T_uv + a__v*b__u*T_vu + a__v*b__v*T_vv'
assert str(T(a,b+c).expand()) == 'a__u*b__u*T_uu + a__u*b__v*T_uv + a__u*c__u*T_uu + a__u*c__v*T_uv + a__v*b__u*T_vu + a__v*b__v*T_vv + a__v*c__u*T_vu + a__v*c__v*T_vv'
assert str(T(a,alpha*b)) == 'a__u*alpha*b__u*T_uu + a__u*alpha*b__v*T_uv + a__v*alpha*b__u*T_vu + a__v*alpha*b__v*T_vv'
#Geometric Derivative With Respect To Slot
assert str(T.pdiff(1)) == '(a_1__u*a_2__u*D{u}T_uu + a_1__u*a_2__v*D{u}T_uv + a_1__v*a_2__u*D{u}T_vu + a_1__v*a_2__v*D{u}T_vv)*e_u + (a_1__u*a_2__u*D{v}T_uu + a_1__u*a_2__v*D{v}T_uv + a_1__v*a_2__u*D{v}T_vu + a_1__v*a_2__v*D{v}T_vv)*e_v/sin(u)**2'
assert str(T.pdiff(2)) == '(a_1__u*a_2__u*D{u}T_uu + a_1__u*a_2__v*D{u}T_uv + a_1__v*a_2__u*D{u}T_vu + a_1__v*a_2__v*D{u}T_vv)*e_u + (a_1__u*a_2__u*D{v}T_uu + a_1__u*a_2__v*D{v}T_uv + a_1__v*a_2__u*D{v}T_vu + a_1__v*a_2__v*D{v}T_vv)*e_v/sin(u)**2'
#Covariant Derivatives
assert str(V.cderiv()) == 'a_2__u*(a_1__u*D{u}V_u + a_1__v*D{u}V_v) + a_2__v*(a_1__u*D{v}V_u + a_1__v*D{v}V_v)'
DT = T.cderiv()
assert str(DT) == 'a_3__u*(a_1__u*a_2__u*D{u}T_uu + a_1__u*a_2__v*D{u}T_uv + a_1__v*a_2__u*D{u}T_vu + a_1__v*a_2__v*D{u}T_vv) + a_3__v*(a_1__u*a_2__u*D{v}T_uu + a_1__u*a_2__v*D{v}T_uv + a_1__v*a_2__u*D{v}T_vu + a_1__v*a_2__v*D{v}T_vv)'
#Define curve on unit sphere manifold
us = Function('u__s')(s)
vs = Function('v__s')(s)
#Define 1-d submanifold on unit shpere manifold
crv1d = sph2d.sm([us,vs],[s])
(es,) = crv1d.mv()
#1-D Manifold On Unit Sphere:
assert str(crv1d.grad) == 'e_s*1/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)*D{s}'
#Define scalar and vector fields on 1-d manifold tangent space
g = crv1d.mv('g','scalar',f=True)
h = crv1d.mv('h','vector',f=True)
assert str(crv1d.grad * g) == 'D{s}g*e_s/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)'
assert str(crv1d.grad | h) == 'D{s}h__s + h__s*sin(u__s)**2*D{s}v__s*D{s}^2v__s/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2) + h__s*sin(2*u__s)*D{s}u__s*D{s}v__s**2/(2*(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)) + h__s*D{s}u__s*D{s}^2u__s/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)'
return
