Is the divergence of a gradient field the trace of the Hessian?

Question

Given a $C^2$ multivariate function $f : \mathbb{R}^d \to \mathbb{R}$, the gradient defines a vector field, the divergence of this vector field, then, should be the trace of the Hessian matrix, right? I'm not entirely sure because the simplified version of divergence is only ever given for $d=3$.

Yes. This is also known as the (flat) Laplacian. – Mikhail Katz Jul 04 '13 at 08:53 — Mikhail Katz, Jul 04 '13 at 08:53

score 1 · Accepted Answer · answered Jul 10 '13 at 05:58

1

yes, evidently, according to @user72694 above, this is known as the "flat" Laplacian

answered Jul 10 '13 at 05:58

fairidox

323

Is the divergence of a gradient field the trace of the Hessian?

1 Answers1