def Test_Reciprocal_Frame():
Print_Function()
Format()
coords = symbols('x y z')
(ex,ey,ez,grad) = MV.setup('e_x e_y e_z',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('\\mbox{Frame}',(eu,ev),'\\mbox{Reciprocal Frame}',(eu_r,ev_r))
print r'%\bm{e}_{u}\cdot\bm{e}^{u} =',(eu|eu_r)
print r'%\bm{e}_{u}\cdot\bm{e}^{v} =',eu|ev_r
print r'%\bm{e}_{v}\cdot\bm{e}^{u} =',ev|eu_r
print r'%\bm{e}_{v}\cdot\bm{e}^{v} =',ev|ev_r
eu = ex+ey+ez
ev = ex-ey
(eu_r,ev_r) = ReciprocalFrame([eu,ev])
oprint('\\mbox{Frame}',(eu,ev),'\\mbox{Reciprocal Frame}',(eu_r,ev_r))
print r'%\bm{e}_{u}\cdot\bm{e}^{u} =',eu|eu_r
print r'%\bm{e}_{u}\cdot\bm{e}^{v} =',eu|ev_r
print r'%\bm{e}_{v}\cdot\bm{e}^{u} =',ev|eu_r
print r'%\bm{e}_{v}\cdot\bm{e}^{v} =',ev|ev_r
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 '\\f{X}{u,v} =',X
MF = Manifold(X,mfvar)
(eu,ev) = MF.Basis()
# Define field on the surface.
g = (v+1)*log(u)
print '\\f{g}{u,v} =',g
# 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 '\\eval{\\nabla g}{u=1,v=0} =', VectorDerivative.subs({u:1,v:0})
# Method 2: Using new Manifold routines.
dg = MF.Grad(g)
print '\\eval{\\f{Grad}{g}}{u=1,v=0} =', dg.subs({u:1,v:0})
dg = MF.grad*g
print '\\eval{\\nabla g}{u=1,v=0} =', dg.subs({u:1,v:0})
return
def Distorted_manifold_with_scalar_function():
Print_Function()
coords = symbols('x y z')
(ex,ey,ez,grad) = MV.setup('e_x e_y e_z',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 '\\eval{dg}{u=1,v=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 'dG =',dG
print 'P(dG) =',MF.Proj(dG)
PS = u*v*eu^ev
print 'P(S) =',PS
print 'dP(S) =',MF.Grad(PS)
print 'P(dP(S)) =',MF.Proj(MF.Grad(PS))
return
def Simple_manifold_with_vector_function_derivative():
Print_Function()
coords = (x,y,z) = symbols('x y z')
basis = (ex, ey, ez, grad) = \
MV.setup('e_x e_y e_z',metric='[1,1,1]',coords=coords)
# Define surface
mfvar = (u,v) = symbols('u v')
X = u*ex+v*ey+(u**2+v**2)*ez
print '\\f{X}{u,v} =',X
MF = Manifold(X,mfvar)
(eu,ev) = MF.Basis()
# Define field on the surface.
g = (v+1)*log(u)
print '\\mbox{Scalar Function: } g =',g
dg = MF.grad*g
dg.Fmt(3,'\\mbox{Scalar Function Derivative: } \\nabla g')
print '\\eval{\\nabla g}{(1,0)} =',dg.subs({u:1,v:0})
# Define vector field on the surface
G = v**2*eu+u**2*ev
print '\\mbox{Vector Function: } G =',G
dG = MF.grad*G
dG.Fmt(3,'\\mbox{Vector Function Derivative: } \\nabla G')
print '\\eval{\\nabla G}{(1,0)} =',dG.subs({u:1,v:0})
return
def Plot_Mobius_Strip_Manifold():
Print_Function()
coords = symbols('x y z')
(ex,ey,ez,grad) = MV.setup('e_x e_y e_z',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