def Distorted_manifold_with_scalar_function():
Print_Function()
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')
X = 2 * u * ex + 2 * v * ey + (u**3 + v**3 / 2) * ez
MF = Manifold(X, mfvar, I=MV.I)
(eu, ev) = MF.Basis()
g = (v + 1) * log(u)
dg = MF.Grad(g)
print('g =', g)
print('dg =', dg)
print('dg(1,0) =', dg.subs({u: 1, v: 0}))
G = u * eu + v * ev
dG = MF.Grad(G)
print('G =', G)
print('P(G) =', MF.Proj(G))
print('zcoef =', simplify(2 * (u**2 + v**2) * (-4 * u**2 - 4 * v**2 - 1)))
print('dG =', dG)
print('P(dG) =', MF.Proj(dG))
PS = u * v * eu ^ ev
print('PS =', PS)
print('dPS =', MF.Grad(PS))
print('P(dPS) =', MF.Proj(MF.Grad(PS)))
return
def Plot_Mobius_Strip_Manifold():
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')
X = (cos(u) + v * cos(u / 2) * cos(u)) * ex + (
sin(u) + v * cos(u / 2) * sin(u)) * ey + v * sin(u / 2) * ez
MF = Manifold(X, mfvar, True, I=MV.I)
MF.Plot2DSurface([0.0, 6.28, 48], [-0.3, 0.3, 12],
surf=False,
skip=[4, 4],
tan=0.15)
return
def Simple_manifold_with_scalar_function_derivative():
coords = (x, y, z) = symbols('x y z')
basis = (e1, e2, e3, grad) = MV.setup('e_1 e_2 e_3',
metric='[1,1,1]',
coords=coords)
# Define surface
mfvar = (u, v) = symbols('u v')
X = u * e1 + v * e2 + (u**2 + v**2) * e3
print X
MF = Manifold(X, mfvar)
# Define field on the surface.
g = (v + 1) * log(u)
# Method 1: Using old Manifold routines.
VectorDerivative = (MF.rbasis[0] / MF.E_sq) * diff(
g, u) + (MF.rbasis[1] / MF.E_sq) * diff(g, v)
print 'Vector derivative =', VectorDerivative.subs({u: 1, v: 0})
# Method 2: Using new Manifold routines.
dg = MF.Grad(g)
print 'Vector derivative =', dg.subs({u: 1, v: 0})
return