1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
|
#!/usr/bin/env python
from __future__ import print_function
from sympy import symbols, log, simplify, diff, cos, sin
from sympy.galgebra import MV, ReciprocalFrame, Format
from sympy.galgebra import oprint
from sympy.galgebra import xdvi, Get_Program, Print_Function
from sympy.galgebra import Manifold
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 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
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_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 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 dummy():
return
def main():
Get_Program()
Test_Reciprocal_Frame()
Distorted_manifold_with_scalar_function()
Simple_manifold_with_scalar_function_derivative()
Simple_manifold_with_vector_function_derivative()
#Plot_Mobius_Strip_Manifold()
xdvi()
return
if __name__ == "__main__":
main()
|
