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| -rw-r--r-- | chapters/2-Manifolds.scm | 190 | ||||
| -rw-r--r-- | log/2022-09-15.md | 28 |
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diff --git a/chapters/2-Manifolds.scm b/chapters/2-Manifolds.scm new file mode 100644 index 0000000..e929510 --- /dev/null +++ b/chapters/2-Manifolds.scm @@ -0,0 +1,190 @@ + +(define R2 (make-manifold R^n 2)) + +(define U (patch 'origin R2)) + +; these are coordinate systems +(define R2-rect (coordinate-system 'rectangular U)) +(define R2-polar (coordinate-system 'polar/cylindrical U)) + +; these are charts, and their inverses +(define R2-rect-chi (chart R2-rect)) +(define R2-rect-chi-inverse (point R2-rect)) +(define R2-polar-chi (chart R2-polar)) +(define R2-polar-chi-inverse (point R2-polar)) + +((compose R2-polar-chi R2-rect-chi-inverse) (up 'x0 'y0)) +; (up +; (sqrt (+ (expt x0 2) (expt y0 2))) ; radius +; (atan y0 x0)) ; angle (theta) + +((compose R2-rect-chi R2-polar-chi-inverse) (up 'r0 'theta0)) +; (up +; (* r0 (cos theta0)) ; x +; (* r0 (sin theta0))) ; y + + +((D (compose R2-rect-chi R2-polar-chi-inverse)) (up 'r0 'theta0)) +; (down +; (up +; (cos theta0) +; (sin theta0)) +; (up +; (* -1 r0 (sin theta0)) +; (* r0 (cos theta0)))) + +(define R2->R (-> (UP Real Real) Real)) +(define f (compose (literal-function 'f-rect R2->R) R2-rect-chi)) + +(define R2-rect-point (R2-rect-chi-inverse (up 'x0 'y0))) + +(define corresponding-polar-point + (R2-polar-chi-inverse + (up (sqrt (+ (square 'x0) (square 'y0))) + (atan 'y0 'x0)))) + +(f R2-rect-point) +; (f-rect (up x0 y0)) + +(f corresponding-polar-point) +; (f-rect (up x0 y0)) + +; confirms that the CAS simplifies to the same point + +(define-coordinates (up x y) R2-rect) +(define-coordinates (up r theta) R2-polar) + +(x (R2-rect-chi-inverse (up 'x0 'y0))) +; x0 + +; expect r0 * cos(theta0) +(x (R2-polar-chi-inverse (up 'r0 'theta0))) +; (* r0 (cos theta0)) + +; expect r0 * sin(theta0) +(y (R2-polar-chi-inverse (up 'r0 'theta0))) +; (* r0 (sin theta0)) + + +;(r (R2-polar-chi-inverse (up 'r0 'theta0))) +; r0 + +; expect sqrt(x0^2 + y0^2) +(r (R2-rect-chi-inverse (up 'x0 'y0))) +; (sqrt (+ (expt x0 2) (expt y0 2))) + +; expect atan(y0, x0) +(theta (R2-rect-chi-inverse (up 'x0 'y0))) +; (atan y0 x0) + +; h: x * r^2 + y^3 +(define h (+ (* x (square r)) (cube y))) + +(h R2-rect-point) +; (+ (expt x0 3) (* x0 (expt y0 2)) (expt y0 3)) +; aka: x0^3 + x0 y0^2 + y0^3 +; x0 (x0^2 + y0^2) + y0^3 + +(h (R2-polar-chi-inverse (up 'r0 'theta0))) +; (+ (* (expt r0 3) (expt (sin theta0) 3)) (* (expt r0 3) (cos theta0))) + + +;;; Exersize 2.1a + +((- r (* 2 'a (+ 1 (cos theta)))) + ((point R2-rect) (up 'x 'y))) +; (/ +; (+ (* -2 a x) +; (* -2 a (sqrt (+ (expt x 2) (expt y 2)))) +; (expt x 2) (expt y 2)) +; (sqrt (+ (expt x 2) (expt y 2)))) + + +; "Lemniscate of Bernoulli" +; (x^2 + y^2)^2 = 2 a^2 (x^2 - y^2) + +((- + (square (+ (square x) (square y))) + (* 2 + (square 'a) + (- (square x) (square y)))) + ((point R2-polar) (up 'r0 'theta0))) +; (+ (* -4 (expt a 2) (expt r0 2) (expt (cos theta0) 2)) +; (* 2 (expt a 2) (expt r0 2)) +; (expt r0 4)) +; +; r^2 / a^2 + 2 = 4 cos^2 (theta) +; => r^2 = 2 a^2 cos(2 theta) + +; this matches Wikipedia, with substitution of a^2 = 2 c^2 (in the rectangular version) + + +;;; Exersize 2.1b + +(define R3-rect-chi (chart R3-rect)) +(define R3-rect-chi-inverse (point R3-rect)) +(define R3-cyl-chi (chart R3-cyl)) +(define R3-cyl-chi-inverse (point R3-cyl)) + +(define-coordinates (up x y z) R3-rect) +(define-coordinates (up r theta z-cyl) R3-cyl) + +; R->R3 +(define (helix-rect t) + (up (* 'R (cos t)) + (* 'R (sin t)) + (* 's t))) + +; R->R3 +(define (helix-cyl t) + (up 'R + t + (* 's t))) + +((compose R3-rect-chi R3-cyl-chi-inverse helix-cyl) 't) +; (up +; (* R (cos t)) +; (* R (sin t)) +; (* s t)) + + +((compose R3-cyl-chi R3-rect-chi-inverse helix-rect) 't) +; (up R +; t +; (* s t)) + +((- helix-rect (compose R3-rect-chi R3-cyl-chi-inverse helix-cyl)) 't) +; (up 0 0 0) + +;;; Exersize 2.2 + +; given polar coordinates of point on plane, get to spherical coordinates of point on sphere with: +((compose + (chart S2-spherical) + (point S2-Riemann) + (chart R2-rect) + (point R2-polar)) + (up 'rho 'theta)) +; (up (acos (/ (+ -1 (expt rho 2)) +; (+ 1 (expt rho 2)))) +; theta) + +; given spherical coordinates of point on sphere, what are the polar +; coordinates of corresponding point on the plane? +((compose + (chart R2-polar) + (point R2-rect) + (chart S2-Riemann) + (point S2-spherical)) + (up 'phi 'lambda)) +; (up (/ (sin phi) +; (+ -1 (cos phi))) +; (atan (* -1 (sin lambda)) +; (* -1 (cos lambda)))) + +; this is just the inverse of the above, right? + +; huh, I would expect the second term to just be lambda. +; atan (-sin(lambda), -cos(lambda)) -> tan^-1(tan(lambda)) -> lambda +; seems like an identity, ok + diff --git a/log/2022-09-15.md b/log/2022-09-15.md new file mode 100644 index 0000000..1dc1008 --- /dev/null +++ b/log/2022-09-15.md @@ -0,0 +1,28 @@ + +## Definitions + +"coordinate patch": an open set of points around a point on a manifold + +"coordinate function" or "chart": continuous function mapping every point in a patch to coordinates (tuples of N real numbers for an N-dimensional manifold) + +multiple charts may be needed to map all points on an entire manifold. a consistent set of charts covering an entire manifold is an "atlas". + + +## Progress + +Went through the second chapter, Manifolds, and did the exersizes. Short and +straight-forward, mostly definitions. Not going in to as much detail/definition +as would if studying Topology directly. + +Started on third chapter, Vector Fields and One Form Fields, and struggling +with terminology a bit. Looking ahead a bit, i'm expecting: + +- Chapters 2,3,4: definitions, geometry, derivatives on manifolds +- Chapters 5,6,7: calculus on manifolds (integration theorems) +- Chapters 7.8: curvature weirdness and the meat/math of general relativity +- Chapters 10,11: physics + + +## TODO + +report typo in scmutils: "Bad point: polar/cylindrial manifold" and "spherical/cylindrial" (should be "cylindrical"?) (in calculus/manifold.scm) |