How to prove the existence of a vector potential for a solenoidal vector field?

Question

If $\textbf{F}$ is a vector field and $\bf{divF}=0$, how would you show that this implies the existence of a vector field $\bf{A}$ such that $\bf{F}=curl A $?

Answer 1

If the domain is star-shaped, $${\mathbf A}({\mathbf x})=\int_0^1{\mathbf F}(t{\mathbf x})\times t{\mathbf x}\,dt.$$ See Vector Potentials.

Answer 2

This follows from the de Rham cohomology group of $\mathbb{R}^3$ being trivial in the second dimension (i.e., every vector field with divergence zero is the curl of another vector field). What is special about $\mathbb{R}^3$ which allows this is that it is contractible to a point, so there are no obstructions to there being such a vector field. (It's also true that every vector field with curl zero is the gradient of some continuous function on $\mathbb{R}^3$ as well.)

The answer Anti-curl operator gives a way of finding the vector field (and since there is a way to find it, this is the existence proof).