polar-laplacian-3d-3.egi

;;;
;;; Spherical coordinates
;;;

(define $x [|r θ φ|])

(define $X [|(* r (sin θ) (cos φ)) ; = x
             (* r (sin θ) (sin φ)) ; = y
             (* r (cos θ))         ; = z
             |])

;;
;; Local coordinates
;;

(define $e ((∂/∂ X~# $) x_#))
e
;[|[| (* (sin θ) (cos φ)) (* (sin θ) (sin φ)) (cos θ) |]
;  [| (* r (cos θ) (cos φ)) (* r (cos θ) (sin φ)) (* -1 r (sin θ)) |]
;  [| (* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ)) 0 |]|]

;;
;; Metric tensor
;;

(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {3 3}))
(define $g~~ (with-symbols {i j} (/ (unit-tensor {3 3})_i_j g_i_j)))

g_#_#;[| [| 1 0 0 |] [| 0 r^2 0 |] [| 0 0 (* r^2 (sin θ)^2) |] |]_#_#
g~#~#;[| [| 1 0 0 |] [| 0 (/ 1 r^2) 0 |] [| 0 0 (/ 1 (* r^2 (sin θ)^2)) |] |]~#~#

;;
;; Laplacian
;;

(define $sqrt-g (sqrt (M.det g_#_#)))
sqrt-g;(* r^2 (sin θ))

(define $Laplacian (/ (contract + (∂/∂ (* sqrt-g (. g~i~j (∂/∂ (f r θ φ) x_j))) x_i)) sqrt-g))
Laplacian
;(/ (+ (* 2 r (sin θ)^2 (f|1 r θ φ)) (* r^2 (sin θ)^2 (f|1|1 r θ φ)) (* (cos θ) (f|2 r θ φ) (sin θ)) (* (sin θ)^2 (f|2|2 r θ φ)) (f|3|3 r θ φ)) (* (sin θ)^2 r^2))