hodge-laplacian-polar.egi

;;; Parameters and metrics

(define $N 2)

(define $x [|r θ|])

(define $g__ [| [| 1 0 |] [| 0 r^2 |] |])
(define $g~~ (M.inverse g_#_#))

;;; Hodge Laplacian

(define $d
  (lambda [%X]
    !((flip ∂/∂) x X)))

(define $hodge
  (lambda [%A]
    (let {[$k (df-order A)]}
      (with-symbols {i j}
        (* (sqrt (abs (M.det g_#_#)))
           (foldl . (. (subrefs A (map 1#j_%1 (between 1 k)))
                       (subrefs (ε' N k) (map 1#i_%1 (between 1 N))))
                  (map 1#g~[i_%1]~[j_%1] (between 1 k))))))))

(define $δ
  (lambda [%A]
    (let {[$r (df-order A)]}
      (* (** -1 (+ (* N r) 1))
         (hodge (d (hodge A)))))))

(define $Δ
  (lambda [%A]
    (match (df-order A) integer
      {[,0 (δ (d A))]
       [,2 (d (δ A))]
       [_ (+ (d (δ A)) (δ (d A)))]})))

(Δ (f r θ))
;(/ (+ (* -1 (f|2|2 r θ)) (* -1 r (f|1 r θ)) (* -1 r^2 (f|1|1 r θ))) r^2)