hodge-laplacian-one-form.egi

;;; Parameters and metrics

(define $N 3)

(define $params [| x y z |])

(define $g__ [| [| 1 0 0 |] [| 0 1 0 |] [| 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 $δ
  (lambda [%A]
    (let {[$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))]
       [,3 (d (δ A))]
       [_ (+ (d (δ A)) (δ (d A)))]})))

(define $ux (function [t x y z]))
(define $uy (function [t x y z]))
(define $uz (function [t x y z]))
(define $u [| ux uy uz |])

(Δ u)
;[| (+ ux|x|x ux|z|z ux|y|y) (+ uy|y|y uy|z|z uy|x|x) (+ uz|z|z uz|y|y uz|x|x) |]

(define $vx (function [t x y z]))
(define $vy (function [t x y z]))
(define $vz (function [t x y z]))
(define $v [|[| 0 vz (* -1 vy) |] [| (* -1 vz) 0 vx |] [| vy (* -1 vx) 0 |]|])

(df-normalize (Δ v))
;[| [| 0 (+ vz|x|x vz|z|z vz|y|y) (+ (* -1 vy|x|x) (* -1 vy|y|y) (* -1 vy|z|z)) |] [| (+ (* -1 vz|y|y) (* -1 vz|x|x) (* -1 vz|z|z)) 0 (+ vx|y|y vx|x|x vx|z|z) |] [| (+ vy|z|z vy|x|x vy|y|y) (+ (* -1 vx|z|z) (* -1 vx|y|y) (* -1 vx|x|x)) 0 |] |]