;;;
;;; Polar coordinates
;;;

(define $x [|r θ|])

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

;;
;; Local coordinates
;;

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

;;
;; Metric tensor
;;

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

g_#_#;[| [| 1 0 |] [| 0 r^2 |] |]_#_#
g~#~#;[| [| 1 0 |] [| 0 (/ 1 r^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 {2 2 2} {0 0 0 (* -1 r) 0 r r 0} )_#_#_#
Γ_1_#_#;[| [| 0 0 |] [| 0 (* -1 r) |] |]_#_#
Γ_2_#_#;[| [| 0 r |] [| r 0 |] |]_#_#

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

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

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

;;
;; Derive Laplacian
;;

(. g~i~j (∂/∂ (∂/∂ (f r θ) x_j) x_i))
;(/ (+ (* (f|1|1 r θ) r^2) (f|2|2 r θ)) r^2)
(. (. g~i~j Γ~k_i_j) (∂/∂ (f r θ) x_k))
;(/ (* -1 (f|1 r θ)) r)

(define $Laplacian (- (. g~i~j (∂/∂ (∂/∂ (f r θ) x_j) x_i))
                        (. (. g~i~j Γ~k_i_j) (∂/∂ (f r θ) x_k))))
Laplacian
;(/ (+ (* (f|1|1 r θ) r^2) (f|2|2 r θ) (* (f|1 r θ) r)) r^2)