Exemplo n.º 1
0
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
Exemplo n.º 2
0
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
Exemplo n.º 3
0
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
Exemplo n.º 4
0
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
Exemplo n.º 5
0
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
Exemplo n.º 6
0
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
Exemplo n.º 7
0
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
Exemplo n.º 8
0
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