blob: 748316b22e8b30bf64f0481927614cfda8fb4cf5 (
plain)
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
|
(define ((vector-field->vector-field-over-map mu:N->M) v-on-M)
(procedure->vector-field
(lambda (f-on-M)
(compose (v-on-M f-on-M) mu:N->M))))
(define (((differential mu) v) f)
(v (compose f mu)))
(define ((form-field->form-field-over-map mu:N->M) w-on-M)
(define (make-fake-vector-field V-over-mu n)
(define ((u f) m)
((V-over-mu f) n))
(procedure->vector-field u))
(procedure->nform-field
(lambda vectors-over-map
(lambda (n)
((apply w-on-M
(map (lambda (V-over-mu)
(make-fake-vector-field V-over-mu n))
vectors-over-map))
(mu:N->M n))))
(get-rank w-on-M)))
(define S2 (make-manifold S^2 2 3))
(define S2-spherical
(coordinate-system-at 'spherical 'north-pole S2))
(define-coordinates (up theta phi) S2-spherical)
(define S2-basis (coordinate-system->basis S2-spherical))
(define-coordinates t R1-rect)
(define mu
(compose (point S2-spherical)
(up (literal-function 'theta)
(literal-function 'phi))
(chart R1-rect)))
(define S2-basis-over-mu
(basis->basis-over-map mu S2-basis))
(define h
(literal-manifold-function 'h-spherical S2-spherical))
(((basis->vector-basis S2-basis-over-mu) h)
((point R1-rect) 't0))
; (down (((partial 0) h-spherical) (up (theta t0) (phi t0)))
; (((partial 1) h-spherical) (up (theta t0) (phi t0))))
(((basis->1form-basis S2-basis-over-mu)
(basis->vector-basis S2-basis-over-mu))
((point R1-rect) 't0))
; (up (down 1 0) (down 0 1))
(((basis->1form-basis S2-basis-over-mu)
((differential mu) d/dt))
((point R1-rect) 't0))
; (up ((D theta) t0) ((D phi) t0))
(define ((pullback-function mu:N->M) f-on-M)
(compose f-on-M mu:N->M))
(define ((pushforward-vector mu:N->M mu^-1:M->N) v-on-N)
(procedure->vector-field
(lambda (f)
(compose (v-on-N (compose f mu:N->M)) mu^-1:M->N))))
(define (pullback-vector-field mu:N->M mu^-1:M->N)
(pushforward-vector mu^-1:M->N mu:N->M))
(define ((pullback-form mu:N->M) omega-on-M)
(let ((k (get-rank omega-on-M)))
(if (= k 0)
((pullback-function mu:N->M) omega-on-M)
(procedure->nform-field
(lambda vectors-on-N
(apply ((form-field->form-field-over-map mu:N->M)
omega-on-M)
(map (differential mu:N_>M) vectors-on-N)))
k))))
(define mu (literal-manifold-map 'MU R2-rect R3-rect))
(define f (literal-manifold-function 'f-rect R3-rect))
(define X (literal-vector-field 'X-rect R2-rect))
(((- ((pullback mu) (d f)) (d ((pullback mu) f))) X)
((point R2-rect) (up 'x0 'y0)))
; 0
(define theta (literal-1form-field 'THETA R3-rect))
(define Y (literal-vector-field 'Y-rect R2-rect))
(((- ((pullback mu) (d theta)) (d ((pullback mu) theta))) X Y)
((point R2-rect) (up 'x0 'y0)))
; 0
;;; Exersize 6.1: Velocities on a Globe
; a)
; b)
|
