#|
;;; Experiments to determine the story

;;; V-on-M is a vector field on M  template  V(f)(m)
;;; V-over-mu is a vector field over map mu 
;;;   mu: N->M, f:M->R  template V(f)(n)
;;; X is a vector field on N  template X(g)(n) where g:N->R
;;; when mu = identity this reduces to the previous covariant-derivative

(define ((vector-over-map->coef mu V-over-mu basis-on-M) n)
  (let ((dual-basis (basis->1form-basis basis-on-M))
	(m (mu n)))
    (s:map/r (lambda (edual)
	       ((edual 
		 (procedure->vector-field
		  (lambda (f) (lambda (m) ((V-over-mu f) n)))
		  'foo))
		m))
	   dual-basis)))

(define ((((covariant-derivative basis-on-M Christoffel-symbols mu:N->M) 
	   X-on-N) 
	  V-over-mu) 
	 f-on-M)
  (let ((vector-basis-on-M (basis->vector-basis basis-on-M)))
    (let ((v-components (vector-over-map->coef mu:N->M V-over-mu basis-on-M))
	  (x-components 
	   (vector-over-map->coef mu:N->M
				  ((differential mu:N->M) X-on-N)
				  basis-on-M)))
      (* (compose (vector-basis-on-M f-on-M) mu:N->M)
	 (+ (X-on-N v-components)
	    (* (* (compose Christoffel-symbols mu:N->M)
		  x-components)
	       v-components))))))
#|
(pe (let ((U (components->vector-field (lambda (x) 1) the-real-line))
	  (mu:N->M (compose (S '->point)
			    (up (literal-function 'theta)
				(literal-function 'phi)))))
      ((vector-over-map->coef 
	mu:N->M
	(((covariant-derivative 2-sphere-basis G-S2-1 mu:N->M)
	  U)
	 ((differential mu:N->M) U))				  
	2-sphere-basis)
       'tau)))
(up (+ (* -1
	  (cos (theta tau))
	  (expt ((D phi) tau) 2)
	  (sin (theta tau)))
       (((expt D 2) theta) tau))
    (+ (/ (* 2
	     ((D theta) tau)
	     (cos (theta tau))
	     ((D phi) tau))
	  (sin (theta tau)))
       (((expt D 2) phi) tau)))
|# |#