Laplacian, Hessian, and Jacobian

All three operators are assembled from the same array of partial derivatives. For a scalar $f$ and vector map $F$,

$$ (\operatorname{Hess}f)_{ij}=\partial_i\partial_j f, \qquad \Delta f=\operatorname{tr}(\operatorname{Hess}f), \qquad J_F=(\partial_jF_i)_{ij}. $$

Coordinates and test fields

We use $f=x^2+y^2+z^2$ and $F=(x^2,y^2,z^2)$. Both have diagonal derivative matrices, which makes the expected answers easy to inspect.

declare symbol x, y, z : MathValue

def parameters : Vector MathValue := [| x, y, z |]
def scalarField : MathValue := x^2 + y^2 + z^2
def vectorField : Vector MathValue := [| x^2, y^2, z^2 |]

∂/∂ scalarField parameters
$\begin{pmatrix} 2 x \\ 2 y \\ 2 z\\ \end{pmatrix}$

Hessian and Laplacian

generateTensor supplies the two free matrix indices. Taking the trace of the resulting Hessian implements the Euclidean Laplacian.

def hessian (f : MathValue) : Matrix MathValue :=
  generateTensor
    (\[a, b] -> ∂/∂ (∂/∂ f parameters_a) parameters_b)
    [3, 3]

def laplacian (f : MathValue) : MathValue := trace (hessian f)

hessian scalarField
$\begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{pmatrix}$
laplacian scalarField
$6$

Jacobian matrix and determinant

The same indexed construction differentiates each component of $F$. Its determinant measures the local volume scaling.

def jacobian (v : Vector MathValue) : Matrix MathValue :=
  generateTensor
    (\[a, b] -> ∂/∂ v_a parameters_b)
    [3, 3]

jacobian vectorField
$\begin{pmatrix} 2 x & 0 & 0 \\ 0 & 2 y & 0 \\ 0 & 0 & 2 z \\ \end{pmatrix}$
M.det (jacobian vectorField)
$8 x y z$

The outputs are $2I_3$, $6$, and $8xyz$, respectively. They show how tensor generation, contraction, and determinant calculation express three familiar multivariable operators in a uniform way.

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