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| -rw-r--r-- | chapters/3-Vector-Fields.scm | 223 | ||||
| -rw-r--r-- | definitions.md | 7 | ||||
| -rw-r--r-- | log/2022-09-20.md | 36 |
3 files changed, 266 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? + diff --git a/definitions.md b/definitions.md new file mode 100644 index 0000000..7b9f419 --- /dev/null +++ b/definitions.md @@ -0,0 +1,7 @@ + +vector field on a manifold + "an assignment of a vector to each point on the manifold" + "In differential geometry, a vector is an operator that takes directional derivatives of manifold functions at its anchor point. The direction and magnitude of the vector are the direction and scale factor of the directional derivative." + "the vector field is an operator that takes a real-valued manifold function and a manifold point and produces a number" + + vector field is a *geometric object*; it is independent of any one coordinate system diff --git a/log/2022-09-20.md b/log/2022-09-20.md new file mode 100644 index 0000000..cede4f2 --- /dev/null +++ b/log/2022-09-20.md @@ -0,0 +1,36 @@ + +## Vector Fields + +Struggling a bit to understand what a "vector field" is. Is it a geometric +object? It is described as an operator, which is sort of confusing. Can it be +considered/visualized separately from any specific manifold function? Are +vector fields one-to-one with manifold functions? + +Suspect this because we are using functional terminology, and trying to define +"vector field" as something with a type signature, aka in terms of what it +"takes" and "returns". + +Remember, a manifold function maps (geometric) points on a manifold to real +numbers. + +Didn't understand the "covariantly"/"contravariantly" bit in the "Coordinate +Transformations" section. + +--- + +Ok, so "vector field on manifold" is a mapping from points on the manifold to +vectors. It is often combined with a manifold function to give the directional +derivative of the function at a point, which is a real number. In the language +of differential geometry, it is an operator that takes a real-valued manifold +function, and a point, and returns a real value. + +A "one-form field" is a geometric operator which works on a vector field to +define a mapping from points to real values. + +Maybe I was over-thinking this? + +TODO: look in Spivak to see definitions there + +--- + +Coordinate-basis one-form fields are "dual" to coordinate-basis vector fields. |