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Diffstat (limited to 'chapters')
| -rw-r--r-- | chapters/5-Integration.scm | 166 |
1 files changed, 166 insertions, 0 deletions
diff --git a/chapters/5-Integration.scm b/chapters/5-Integration.scm new file mode 100644 index 0000000..ab81784 --- /dev/null +++ b/chapters/5-Integration.scm @@ -0,0 +1,166 @@ + +; deps +(define R2->R (-> (UP Real Real) Real)) +(define R2-rect-chi (chart R2-rect)) +(define R2-rect-chi-inverse (point R2-rect)) +(define R2-rect-point (R2-rect-chi-inverse (up 'x0 'y0))) + +(define R3->R (-> (UP Real Real Real) Real)) +(define R3-rect-chi (chart R3-rect)) +(define R3-rect-chi-inverse (point R3-rect)) +(define R3-rect-point (R3-rect-chi-inverse (up 'x0 'y0 'z0))) + +(define-coordinates (up x y z) R3-rect) + +; two regular old vectors +(define u (+ (* 'u^0 d/dx) (* 'u^1 d/dy))) +(define v (+ (* 'v^0 d/dx) (* 'v^1 d/dy))) + +(((wedge dx dy) u v) R3-rect-point) +; (+ (* u^0 v^1) +; (* -1 u^1 v^0)) + +(define-coordinates (up r theta z) R3-cyl) + +; two more vectors +(define a (+ (* 'a^0 d/dr) (* 'a^1 d/dtheta))) +(define b (+ (* 'b^0 d/dr) (* 'b^1 d/dtheta))) + +(((wedge dr dtheta) a b) ((point R3-cyl) (up 'r0 'theta0 'z0))) +; (+ (* a^0 b^1) (* -1 a^1 b^0)) + +; the same thing, as expected + +; fancy verification of determinant +; note that we are re-defining u,v + +(define u (+ (* 'u^0 d/dx) (* 'u^1 d/dy) (* 'u^2 d/dz))) +(define v (+ (* 'v^0 d/dx) (* 'v^1 d/dy) (* 'v^2 d/dz))) +(define w (+ (* 'w^0 d/dx) (* 'w^1 d/dy) (* 'w^2 d/dz))) + +(((wedge dx dy dz) u v w) R3-rect-point) +; (+ (* u^0 v^1 w^2) +; (* -1 u^0 v^2 w^1) +; (* -1 u^1 v^0 w^2) +; (* u^1 v^2 w^0) +; (* u^2 v^0 w^1) +; (* -1 u^2 v^1 w^0)) + +(- (((wedge dx dy dz) u v w) R3-rect-point) + (determinant + (matrix-by-rows (list 'u^0 'u^1 'u^2) + (list 'v^0 'v^1 'v^2) + (list 'w^0 'w^1 'w^2)))) +; 0 + +; continuing with bits after exersize 5.1... + +(define a (literal-manifold-function 'alpha R3-rect)) +(define b (literal-manifold-function 'beta R3-rect)) +(define c (literal-manifold-function 'gamma R3-rect)) + +(define theta (+ (* a dx) (* b dy) (* c dz))) + +(define X (literal-vector-field 'X-rect R3-rect)) +(define Y (literal-vector-field 'Y-rect R3-rect)) + +(((- (d theta) + (+ (wedge (d a) dx) + (wedge (d b) dy) + (wedge (d c) dz))) + X Y) + R3-rect-point) +; 0 + +(define omega + (+ (* a (wedge dy dz)) + (* b (wedge dz dx)) + (* c (wedge dx dy)))) + +(define Z (literal-vector-field 'Z-rect R3-rect)) + +(((- (d omega) + (+ (wedge (d a) dy dz) + (wedge (d b) dz dx) + (wedge (d c) dx dy))) + X Y Z) + R3-rect-point) +; 0 + +(define v (literal-vector-field 'v-rect R2-rect)) +(define w (literal-vector-field 'w-rect R2-rect)) + +(define R2-rect-basis (coordinate-system->basis R2-rect)) + +(define alpha (literal-function 'alpha R2->R)) +(define beta (literal-function 'beta R2->R)) + +(let ((dx (ref (basis->1form-basis R2-rect-basis) 0)) + (dy (ref (basis->1form-basis R2-rect-basis) 1))) + (((- (d (+ (* (compose alpha (chart R2-rect)) dx) + (* (compose beta (chart R2-rect)) dy))) + (* (compose (- ((partial 0) beta) + ((partial 1) alpha)) + (chart R2-rect)) + (wedge dx dy))) + v w) + R2-rect-point)) +; 0 + +(define a (literal-manifold-function 'a-rect R3-rect)) +(define b (literal-manifold-function 'b-rect R3-rect)) +(define c (literal-manifold-function 'c-rect R3-rect)) + +(define flux-through-boundary-element + (+ (* a (wedge dy dz)) + (* b (wedge dz dx)) + (* c (wedge dx dy)))) + +(define production-in-volume-element + (* (+ (d/dx a) (d/dy b) (d/dz c)) + (wedge dx dy dz))) + +(define X (literal-vector-field 'X-rect R3-rect)) +(define Y (literal-vector-field 'Y-rect R3-rect)) +(define Z (literal-vector-field 'Z-rect R3-rect)) + +(((- production-in-volume-element + (d flux-through-boundary-element)) + X Y Z) + R3-rect-point) +; 0 + +;;; Exercise 5.1: Wedge Product + +; let's do R2-polar! small and simple + +(define-coordinates (up r theta) R2-polar) +(define R2-polar-chi (chart R2-polar)) +(define R2-polar-chi-inverse (point R2-polar)) +(define R2-polar-point (R2-polar-chi-inverse (up 'r0 'theta0))) + +; a) verify that the wedge product is associative for forms in R2-polar + +; A wedge ( B wedge C ) == ( A wedge B) wedge C + +(define A (literal-1form-field 'a R2-polar)) +(define B (literal-1form-field 'b R2-polar)) +(define C (literal-1form-field 'c R2-polar)) + +(define m (+ (* 'm^0 d/dr) (* 'm^1 d/dtheta))) +(define n (+ (* 'n^0 d/dr) (* 'n^1 d/dtheta))) +(define p (+ (* 'p^0 d/dr) (* 'p^1 d/dtheta))) + + +(((- (wedge (wedge A B) C) + (wedge A (wedge B C))) m n p) R2-polar-point) +; 0 + +; b) verify that formula (5.17) is true in R2-polar +; (dx wedge dy wedge ...) (d/dx, d/dy, ...) = 1 + +(((wedge dr dtheta) d/dr d/dtheta) R2-polar-point) +; 1 + +; exersize 5.2 and 5.3 would be in notebook, but not planning to do these right +; now |