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Diffstat (limited to 'chapters')
| -rw-r--r-- | chapters/3-Vector-Fields.scm | 223 |
1 files changed, 223 insertions, 0 deletions
diff --git a/chapters/3-Vector-Fields.scm b/chapters/3-Vector-Fields.scm new file mode 100644 index 0000000..2e15c18 --- /dev/null +++ b/chapters/3-Vector-Fields.scm @@ -0,0 +1,223 @@ + +; pre-reqs +(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 (components->vector-field components coordsys) + (define (v f) + (compose (* (D (compose f (point coordsys))) + components) + (chart coordsys))) + (procedure->vector-field v)) + +(define v + (components->vector-field + (up (literal-function 'b^0 R2->R) + (literal-function 'b^1 R2->R)) + R2-rect)) + +((v (literal-manifold-function 'f-rect R2-rect)) R2-rect-point) +; (+ (* (b^0 (up x0 y0)) +; (((partial 0) f-rect) (up x0 y0))) +; (* (b^1 (up x0 y0)) +; (((partial 1) f-rect) (up x0 y0)))) + +((v (chart R2-rect)) R2-rect-point) +; (up (b^0 (up x0 y0)) (b^1 (up x0 y0))) + +(define (coordinatize v coordsys) + (define ((coordinatized-v f) x) + (let ((b (compose (v (chart coordsys)) + (point coordsys)))) + (* ((D f) x) (b x)))) + (make-operator coordinatized-v)) + +(((coordinatize v R2-rect) (literal-function 'f-rect R2->R)) + (up 'x0 'y0)) +; (+ (* (b^0 (up x0 y0)) +; (((partial 0) f-rect) (up x0 y0))) +; (* (b^1 (up x0 y0)) +; (((partial 1) f-rect) (up x0 y0)))) + +(define-coordinates (up x y) R2-rect) +(define-coordinates (up r theta) R2-polar) + +((d/dx (square r)) R2-rect-point) +; (* 2 x0) + +(((+ d/dx (* 2 d/dy)) (+ (square r) (* 3 x))) R2-rect-point) +; (+ 3 (* 2 x0) (* 4 y0)) + +(define circular (- (* x d/dy) (* y d/dx))) + +(series:for-each print-expression + (((exp (* 't circular)) (chart R2-rect)) + ((point R2-rect) (up 1 0))) + 6) +; (up 0 t) +; (up (* -1/2 (expt t 2)) 0) +; (up 0 (* -1/6 (expt t 3))) +; (up (* 1/24 (expt t 4)) 0) +; (up 0 (* 1/120 (expt t 5))) + +(define ((((evolution order) delta-t v) f) m) + (series:sum + (((exp (* delta-t v)) f) m) + order)) + +((((evolution 6) 'delta-t circular) (chart R2-rect)) + ((point R2-rect) (up 1 0))) +; (up (+ 1 +; (* -1/720 (expt delta-t 6)) +; (* 1/24 (expt delta-t 4)) +; (* -1/2 (expt delta-t 2))) +; (+ (* 1/120 (expt delta-t 5)) +; (* -1/6 (expt delta-t 3)) +; delta-t)) + +; "note: these are jus tthe series expansion for cos(delta-t) and sin(delta-t)" + +;;; Exercise 3.1 + +(print-expression "==== Exercise 3.1") +(define R5 (make-manifold R^n 5)) +(define R5-rect (coordinate-system-at 'rectangular 'origin R5)) +(define R5->R (-> (UP Real Real Real Real Real) Real)) + +(define-coordinates (up pt px py pvx pvy) R5-rect) + +; this isn't really true... it is more of a function of two coordinates. but +; could consider it a manifold function ignoring most? hrm. +(define Ax (literal-manifold-function 'Ax R5-rect)) +(define Ay (literal-manifold-function 'Ay R5-rect)) + +;(define v-newton-planar +; (components->vector-field +; (up 1 +; pvx +; pvy +; Ax +; Ay) +; R5-rect)) + +(define v-newton-planar + (+ (* 1 d/dpt) + (* pvx d/dpx) + (* pvy d/dpy) + (* Ax d/dpvx) + (* Ay d/dpvy))) + +(series:for-each print-expression + (((exp (* 't v-newton-planar)) (chart R5-rect)) + ((point R5-rect) (up 't0 'px0 'py0 'pvx0 'pvy0))) + 3) +;(up t0 px0 py0 pvx0 pvy0) +;(up t (* pvx0 t) (* pvy0 t) (* t (Ax (up t0 px0 py0 pvx0 pvy0))) (* t (Ay (up t0 px0 py0 pvx0 pvy0)))) +;(up +; 0 +; (* 1/2 (expt t 2) (Ax (up t0 px0 py0 pvx0 pvy0))) +; (* 1/2 (expt t 2) (Ay (up t0 px0 py0 pvx0 pvy0))) +; (+ (* 1/2 pvx0 (expt t 2) (((partial 1) Ax) (up t0 px0 py0 pvx0 pvy0))) +; (* 1/2 pvy0 (expt t 2) (((partial 2) Ax) (up t0 px0 py0 pvx0 pvy0))) +; (* 1/2 (expt t 2) (Ay (up t0 px0 py0 pvx0 pvy0)) (((partial 4) Ax) (up t0 px0 py0 pvx0 pvy0))) +; (* 1/2 (expt t 2) (Ax (up t0 px0 py0 pvx0 pvy0)) (((partial 3) Ax) (up t0 px0 py0 pvx0 pvy0))) +; (* 1/2 (expt t 2) (((partial 0) Ax) (up t0 px0 py0 pvx0 pvy0)))) +; (+ (* 1/2 pvx0 (expt t 2) (((partial 1) Ay) (up t0 px0 py0 pvx0 pvy0))) +; (* 1/2 pvy0 (expt t 2) (((partial 2) Ay) (up t0 px0 py0 pvx0 pvy0))) +; (* 1/2 (expt t 2) (Ay (up t0 px0 py0 pvx0 pvy0)) (((partial 4) Ay) (up t0 px0 py0 pvx0 pvy0))) +; (* 1/2 (expt t 2) (Ax (up t0 px0 py0 pvx0 pvy0)) (((partial 3) Ay) (up t0 px0 py0 pvx0 pvy0))) +; (* 1/2 (expt t 2) (((partial 0) Ay) (up t0 px0 py0 pvx0 pvy0))))) + +; this isn't really complete/correct, though it is sort of close. hrm. + +;;; One-form Fields + +(define omega + (components->1form-field + (down (literal-function 'a_0 R2->R) + (literal-function 'a_1 R2->R)) + R2-rect)) + +((omega (down d/dx d/dy)) R2-rect-point) +; (down (a_0 (up x0 y0)) (a_1 (up x0 y0))) + +(define omega (literal-1form-field 'a R2-rect)) + +(((d (literal-manifold-function 'f-rect R2-rect)) + (coordinate-system->vector-basis R2-rect)) + R2-rect-point) + +; (down (((partial 0) f-rect) (up x0 y0)) +; (((partial 1) f-rect) (up x0 y0))) + +(((d (literal-manifold-function 'f-rect R2-polar)) + (coordinate-system->vector-basis R2-rect)) + ((point R2-polar) (up 'r 'theta))) +; (down (/ (+ (* r (cos theta) (((partial 0) f-rect) (up r theta))) (* -1 (sin theta) (((partial 1) f-rect) (up r theta)))) r) +; (/ (+ (* r (sin theta) (((partial 0) f-rect) (up r theta))) (* (cos theta) (((partial 1) f-rect) (up r theta)))) r)) + +(define-coordinates (up x y) R2-rect) + +((dx d/dy) R2-rect-point) +; 0 + +((dx d/dx) R2-rect-point) +; 1 + +((dx circular) R2-rect-point) +; (* -1 y0) + + +((dy circular) R2-rect-point) +; x0 + +((dr circular) R2-rect-point) +; 0 + +((dtheta circular) R2-rect-point) +; 1 + +(define f (literal-manifold-function 'f-rect R2-rect)) + +(((- circular d/dtheta) f) R2-rect-point) +; 0 + +(define omega (literal-1form-field 'a R2-rect)) + +(define v (literal-vector-field 'b R2-rect)) + +((omega v) R2-rect-point) +; (+ (* (b^0 (up x0 y0)) +; (a_0 (up x0 y0))) +; (* (b^1 (up x0 y0)) +; (a_1 (up x0 y0)))) + +;;; Exercise 3.2 + +; not done + +;;; Exersize 3.3 Hill Climbing + +(print-expression "==== Exercise 3.3") + +(define S2-spherical-point ((point S2-spherical) (up 'theta0 'phi0))) + +(define h (literal-manifold-function 'h-spherical S2-spherical)) +(define v-walk (literal-vector-field 'v-walk S2-spherical)) + +(define (power mass) + (* mass (v-walk h))) + +((power 'mass0) S2-spherical-point) +; (+ (* mass0 +; (v-walk^0 (up theta0 phi0)) +; (((partial 0) h-spherical) (up theta0 phi0))) +; (* mass0 +; (v-walk^1 (up theta0 phi0)) +; (((partial 1) h-spherical) (up theta0 phi0)))) + +; I think I got this correct? + |