hodge-laplacian-E3.egi

;;; Parameters and metrics

(define $N 2)

(define $params [|x y|])

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

;;; Hodge Laplacian

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

(define $hodge
  (lambda [%A]
    (let {[$k (df-order A)]}
      (with-symbols {i j}
        (* (sqrt (abs (M.det g_#_#)))
           (foldl . (. (ε' N k)_[i_1]..._[i_N]
                       A..._[j_1]..._[j_k])
                  (map 1#g~[i_%1]~[j_%1] (between 1 k))))))))

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

(define $f (function [x y]))

(d f)
;[| f|x f|y |]

(hodge (d f))
;[| (* -1 f|y) f|x |]

(d (hodge (d f)))
;[| [| (* -1 f|y|x) f|x|x |] [| (* -1 f|y|y) f|x|y |] |]

(hodge (d (hodge (d f))))
;(+ f|y|y f|x|x)

(Δ f)
;(+ (* -1 f|y|y) (* -1 f|x|x))