Irrotational fields and divergence

Question

Let $F,G$ be $C^1$ vector fields from $\mathbb R^n$ in itself. The condition $$\int_{\partial A}F\cdot \nu_A\ d\sigma=\int_{\partial A} G\cdot \nu_A\ d\sigma$$ for every bounded domain $A$ whose boundery is $C^1$ does not imply that $F-G$ is constant.

Prove that if $F$ and $G$ are irrotational and bounded then $F-G$ must be constant.

Until now I've obtained that $\int_A div (F-G)=0$, so $div (F-G)=0$ (correct?).

General suggestions about properties of irrotational and zero-divergence vector fields are welcome too.

Thank you in advance!

Answer 1

With the notation $H=F-G$ we have

1. $H$ is curl free and divergence free $\implies$ $H=\nabla \phi$ where $\phi$ is a harmonic function in $\Bbb R^n$.
2. Partial derivatives of a harmonic function are harmonic functions (partial derivation and Laplace operator commute as harmonic functions are sufficiently smooth), hence, the components of $H$ are harmonic too. They are also bounded in $\Bbb R^n$ by the assumption.
3. Liouville's theorem says now that the components of $H$ must be constant functions.