Util for converting between Cartesian coordinates and spherical coordinates(with the physics symbolic convention).

spherical.scm

(define (xyz->ρ x y z)
  (sqrt (+
         (expt x 2)
         (expt y 2)
         (expt z 2))))

(define (zρ->θ z ρ)
  (acos (/ z ρ)))

(define (yx->φ y x)
  (atan (/ y x)))

(define (xyz->ρθφ x y z)
  (define (ρ) (xyz->ρ x y z))
  (define (θ) (zρ->θ z (ρ)))
  (define (φ) (yx->φ y x))
  (list (ρ) (θ) (φ)))

(define (ρθφ->x ρ θ φ)
  (*
   ρ
   (sin θ)
   (cos φ)))

(define (ρθφ->y ρ θ φ)
  (*
   ρ
   (sin θ)
   (sin φ)))

(define (ρθ->z ρ θ)
  (* ρ (cos θ)))

(define (ρθφ->xyz ρ θ φ)
  (define (x) (ρθφ->x ρ θ φ))
  (define (y) (ρθφ->y ρ θ φ))
  (define (z) (ρθ->z ρ θ))
  (list (x) (y) (z)))