Functional Differential Geometry

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(define R2 (make-manifold R^n 2))

(define U (patch 'origin R2))


(define R2-rect (coordinate-system 'rectangular U))

(define R2-polar (coordinate-system 'polar/cylindrical U))


(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))


(show-expression
 ((compose R2-polar-chi R2-rect-chi-inverse)
  (up 'x_0 'y_0)))

(show-expression
 ((D (compose R2-rect-chi R2-polar-chi-inverse))
  (up 'r_0 'theta_0)))

(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 'x_0 'y_0)))

(define corresponding-polar-point
  (R2-polar-chi-inverse
   (up (sqrt (+ (square 'x_0) (square 'y_0)))
       (atan 'y_0 'x_0))))


(f R2-rect-point)

(f corresponding-polar-point)

(show-expression
 (f R2-rect-point))

(show-expression
 (f corresponding-polar-point))

— Me@2022-11-18 11:22:36 AM

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2022.11.18 Friday (c) All rights reserved by ACHK