def Test_Reciprocal_Frame():
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')
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)
return
def Test_Reciprocal_Frame():
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')
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
return
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 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():
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 = (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 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():
Print_Function()
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
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