Show that every solution $G$ of the equation curl $(G) = F$ is of the form $G = G_0+\nabla f$.

Question

Let $F$ be a solenoid field, that is, div $F = 0$, in a simply connected domain $D \leq \mathbb{R}^3$ and let $G_0$ be a potential field of $F$, that is, curl $(G_0) = F$. Show that every solution $G$ of the equation curl $(G) = F$ is of the form $G = G_0+\nabla f$, where $f$ is a scalar field.

Using that div $F = 0$, I have to div curl $(G_0) = 0$, but I don't know if that helps me.

Any idea?

Answer 1

Hint: Take a solution $G$, and consider the field $G-G_0$. What is the curl of this field? What does that mean?