; 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?