If $\vec{\nabla}× \vec{A}$=0, does this qualify as the necessary and sufficient condition for being able to write $\vec{A}=\vec{\nabla}$f?

Question

If $\vec{\nabla} × \vec{A}=\vec{0}$, does this qualify as the necessary and sufficient condition for being able to writ $\vec{A}=\vec{\nabla}$ f where f is any scalar function?

Answer 1

Necessary, yes, which is quite easy to prove.

If you're working with two or three dimensional vector fields, the condition is also sufficient, but not in any other dimension (but that begs the question how the cross product is defined in other dimensions).

The keyword you're looking for is Conservative Vector field.

PS: If you're just working on the subset of $\mathbb{R}^3$, make sure that it is simply connected!