hodge-laplacian.egi

;;; Parameters and metrics

(define $N 2)

(define $params [|x y|])

(define $g__ [| [| (G_1_1 x y) (G_1_2 x y) |] [| (G_2_1 x y) (G_2_2 x y) |] |])
(define $g~~ [| [| (G~1~1 x y) (G~1~2 x y) |] [| (G~2~1 x y) (G~2~2 x y) |] |])

;;; 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 . (. (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)))]})))

(d (f x y))
(hodge (d (f x y)))
(d (hodge (d (f x y))))
(δ (d (f x y)))
(Δ (f x y))
;