Necessary and sufficient for vector field to be conservative

Question

I found the following statement in this youtube lecture:

Let $A\subset \mathbb{R}^n$ be open and convex and let $f:A\to\mathbb{R}^n$ be continuously differentiable vector field.

Then $f$ is conservative iff the Jacobian of $f$ is symmetric on $A$.

I couldn't find a citable reference for this (for $n\ne3$).

Does someone have one?

Update: I'm still looking for a citable reference.

Answer 1

$f$ conservative means exist an scalar field $\phi$ with $f = \nabla\phi$. Apply now Schwarz's theorem. For the reverse implication see my (a bit more general) answer in Poincaré lemma for star shaped domain.

Reference: Fundamentals of Differential Geometry by Serge Lang. Theorem 4.1 of chap. V.