laplacian-hessian-jacobian.egi

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

(define $∂ (∂/∂ $ parameters))

(∂_i [| x^2 y^2 z^2 |]_i)
;[| (* 2 x) (* 2 y) (* 2 z) |]_i

(∂_i [| x^2 y^2 z^2 |]_j)
;[| [| (* 2 x) 0 0 |] [| 0 (* 2 y) 0 |] [| 0 0 (* 2 z) |] |]_i_j

(define $Δ
  (lambda [%f]
    (with-symbols {i}
      (contract + (∂~i (∂_i f))))))

(define $Hessian
  (lambda [%f]
    (with-symbols {i j}
      (∂_i (∂_j f)))))

(define $Jacobian
  (lambda [%v]
    (with-symbols {i j}
      (M.det (∂_i v_j)))))

(Δ (+ x^2 y^2 z^2))
;6

(Hessian (+ x^2 y^2 z^2))
;[| [| 2 0 0 |] [| 0 2 0 |] [| 0 0 2 |] |]

(Jacobian [| x^2 y^2 z^2 |])
;(* 8 x y z)