; 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-polar-chi (chart R2-polar))
(define R2-polar-chi-inverse (point R2-polar))
(define R2-rect-point (R2-rect-chi-inverse (up 'x0 'y0)))
(define R2-polar-point (R2-polar-chi-inverse (up 'r0 'theta0)))

(define-coordinates (up x y) R2-rect)
(define-coordinates (up r theta) R2-polar)

(define e0
  (+ (* (literal-manifold-function 'e0x R2-rect) d/dx)
     (* (literal-manifold-function 'e0y R2-rect) d/dy)))

(define e1
  (+ (* (literal-manifold-function 'e1x R2-rect) d/dx)
     (* (literal-manifold-function 'e1y R2-rect) d/dy)))

(define e-vector-basis (down e0 e1))
(define e-dual-basis
  (vector-basis->dual e-vector-basis R2-polar))

((e-dual-basis e-vector-basis) R2-rect-point)
; (up (down 1 0) (down 0 1))

((e-dual-basis e-vector-basis) R2-polar-point)
; (up (down 1 0) (down 0 1))

; remember, e-vector-basis is "down"
(define v
  (* (up (literal-manifold-function 'b^0 R2-rect)
         (literal-manifold-function 'b^1 R2-rect))
     e-vector-basis))

((e-dual-basis v) R2-rect-point)
; (up (b^0 (up x0 y0)) (b^1 (up x0 y0)))

(define (Jacobian to-basis from-basis)
  (s:map/r (basis->1form-basis to-basis)
           (basis->vector-basis from-basis)))

(define b-rect
  ((coordinate-system->1form-basis R2-rect)
   (literal-vector-field 'b R2-rect)))

(define b-polar
  (* (Jacobian (coordinate-system->basis R2-polar)
               (coordinate-system->basis R2-rect))
     b-rect))

(b-rect ((point R2-rect) (up 'x0 'y0)))
; (up (b^0 (up x0 y0)) (b^1 (up x0 y0)))

(b-polar ((point R2-rect) (up 'x0 'y0)))
; (up (/ (+ (* x0 (b^0 (up x0 y0))) (* y0 (b^1 (up x0 y0))))
;        (sqrt (+ (expt x0 2) (expt y0 2))))
;     (/ (+ (* x0 (b^1 (up x0 y0))) (* -1 y0 (b^0 (up x0 y0))))
;        (+ (expt x0 2) (expt y0 2))))
; same as...
; (x0 b0(m) + y0 b1(m)) / r
; (x0 b1(m) - y0 b0(m)) / r^2

; Commutators

(let* ((polar-basis (coordinate-system->basis R2-polar))
       (polar-vector-basis (basis->vector-basis polar-basis))
       (polar-dual-basis (basis->1form-basis polar-basis))
       (f (literal-manifold-function 'f-rect R2-rect)))
  ((- ((commutator e0 e1) f)
      (* (- (e0 (polar-dual-basis e1))
            (e1 (polar-dual-basis e0)))
         (polar-vector-basis f)))
   R2-rect-point))
; 0

(define-coordinates (up x y z) R3-rect)

(define R3->R (-> (UP Real Real Real) Real))
(define R3-rect-chi (chart R3-rect))
(define R3-rect-chi-inverse (point R3-rect))
(define R3-rect-point (R3-rect-chi-inverse (up 'x0 'y0 'z0)))

(define Jz (- (* x d/dy) (* y d/dx)))
(define Jx (- (* y d/dz) (* z d/dy)))
(define Jy (- (* z d/dx) (* x d/dz)))

; huh, 'g' here was not defined. is it supposed to be a metric? probably not. a function?
;(((+ (commutator Jx Jy) Jz) g) R3-rect-point)
(define g (compose (literal-function 'g-rect R3->R) R3-rect-chi))

; oh, I see, there was a footnote. I defined things a little differently, could have been:

; (define R3-rect (coordinate-system-at 'rectangular 'origin R3))
; (define-coordinates (up x y z) R3-rect)
; (define R3-rect-point ((point R3-rect) (up 'x0 'y0 'z0)))
; (define g (literal-manifold-function 'g-rect R3-rect))


(((+ (commutator Jx Jy) Jz) g) R3-rect-point)
; 0
(((+ (commutator Jy Jz) Jx) g) R3-rect-point)
; 0
(((+ (commutator Jz Jx) Jy) g) R3-rect-point)
; 0

(define Euler-angles (coordinate-system-at 'Euler 'Euler-patch SO3))
(define Euler-angles-chi-inverse (point Euler-angles))
(define-coordinates (up theta phi psi) Euler-angles)
(define SO3-point ((point Euler-angles) (up 'theta 'phi' psi)))
(define f (literal-manifold-function 'f-Euler Euler-angles))

; from p48
(define e_x (+ (* (cos phi) d/dtheta)
               (* -1 (sin phi) (/ (cos theta) (sin theta)) d/dphi)
               (* (/ (sin phi) (sin theta)) d/dpsi)))
(define e_y (+ (* (cos phi) (/ (cos theta) (sin theta)) d/dphi)
               (* (sin phi) d/dtheta)
               (* -1 (/ (cos phi) (sin theta)) d/dpsi)))
(define e_z d/dphi)

(((+ (commutator e_x e_y) e_z) f) SO3-point)
; 0

(((+ (commutator e_y e_z) e_x) f) SO3-point)
; 0

(((+ (commutator e_z e_x) e_y) f) SO3-point)
; 0

; that is a fair amount of computer algebra! verified by fiddling with the
; definition of e_x, and the subtraction no longer comes out to zero

;;; Exercise 4.1 Alternate Angles

; 4.1a


;;; Exersize 4.2 General Commutators

; verify equation (4.38) (not done)

;;; Exersize 4.3 SO(3) Basis and Angular Momentum Basis


((Jx g) R3-rect-point)

((e_x f) SO3-point)