polar-laplacian-3d-2.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)) |] |]~#~#

;;
;; Christoffel symbols of the first kind
;;

(define $Γ___
  (with-symbols {j k l}
    (* (/ 1 2)
       (+ (∂/∂ g_j_l x~k)
          (∂/∂ g_j_k x~l)
          (* -1 (∂/∂ g_k_l x~j))))))

Γ_#_#_#;(tensor {3 3 3} {0 0 0 0 (* -1 r) 0 0 0 (* -1 r (sin θ)^2) 0 r 0 r 0 0 0 0 (* -1 r^2 (sin θ) (cos θ)) 0 0 (* r (sin θ)^2) 0 0 (* r^2 (sin θ) (cos θ)) (* r (sin θ)^2) (* r^2 (sin θ) (cos θ)) 0} )_#_#_#
Γ_1_#_#;[| [| 0 0 0 |] [| 0 (* -1 r) 0 |] [| 0 0 (* -1 r (sin θ)^2) |] |]_#_#
Γ_2_#_#;[| [| 0 r 0 |] [| r 0 0 |] [| 0 0 (* -1 r^2 (sin θ) (cos θ)) |] |]_#_#
Γ_3_#_#;[| [| 0 0 (* r (sin θ)^2) |] [| 0 0 (* r^2 (sin θ) (cos θ)) |] [| (* r (sin θ)^2) (* r^2 (sin θ) (cos θ)) 0 |] |]_#_#

;;
;; Christoffel symbols of the second kind
;;

(define $Γ~__
  (with-symbols {i j k l}
    (. g~i~j Γ_j_k_l)))

Γ~#_#_#;(tensor {3 3 3} {0 0 0 0 (* -1 r) 0 0 0 (* -1 r (sin θ)^2) 0 (/ 1 r) 0 (/ 1 r) 0 0 0 0 (* -1 (sin θ) (cos θ)) 0 0 (/ 1 r) 0 0 (/ (cos θ) (sin θ)) (/ 1 r) (/ (cos θ) (sin θ)) 0} )~#_#_#
Γ~1_#_#;[| [| 0 0 0 |] [| 0 (* -1 r) 0 |] [| 0 0 (* -1 r (sin θ)^2) |] |]_#_#
Γ~2_#_#;[| [| 0 (/ 1 r) 0 |] [| (/ 1 r) 0 0 |] [| 0 0 (* -1 (sin θ) (cos θ)) |] |]_#_#
Γ~3_#_#;[| [| 0 0 (/ 1 r) |] [| 0 0 (/ (cos θ) (sin θ)) |] [| (/ 1 r) (/ (cos θ) (sin θ)) 0 |] |]_#_#

;;
;; Laplacian
;;

(. g~i~j (∂/∂ (∂/∂ (f r θ φ) x~j) x~i))
;(/ (+ (* (f|1|1 r θ φ) r^2 (sin θ)^2) (* (f|2|2 r θ φ) (sin θ)^2) (f|3|3 r θ φ)) (* r^2 (sin θ)^2))
(. (. g~i~j Γ~k_i_j) (∂/∂ (f r θ φ) x~k))
;(/ (+ (* -2 (f|1 r θ φ) r (sin θ)) (* -1 (cos θ) (f|2 r θ φ))) (* r^2 (sin θ)))

(define $Laplacian (- (. g~i~j (∂/∂ (∂/∂ (f r θ φ) x~j) x~i))
                        (. (. g~i~j Γ~k_i_j) (∂/∂ (f r θ φ) x~k))))
Laplacian
;(/ (+ (* (f|1|1 r θ φ) r^2 (sin θ)^2) (* (f|2|2 r θ φ) (sin θ)^2) (f|3|3 r θ φ) (* 2 (f|1 r θ φ) r (sin θ)^2) (* (cos θ) (f|2 r θ φ) (sin θ))) (* r^2 (sin θ)^2))