When would $\underline{\nabla} \cdot \underline{F} = 0$?

Question

I see that $$\underline{\nabla} \cdot \underline{F} = \frac{\partial F_1}{\partial x_1} + \frac{\partial F_2}{\partial x_2} + \frac{\partial F_3}{\partial x_3} = \frac{\partial F_i}{\partial x_i}$$ But what would be the case for this to be zero?

Answer 1

Such a vector field is called divergenceless. A physical example from fluid dynamics would involve the velocity field of the fluid. This is divergenceless if there are no sources and sinks (assuming an incompressible fluid).

An important theorem for divergenceless is the Gauss theorem which in this context implies that $$\int_S \vec{F}\cdot \vec{n} \,dS =0$$ with $S$ an arbitrary closed surface and $\vec{n}$ the normal to it.

An trivial example would be $$\vec{F} = \begin{pmatrix} x_2 \\ x_1\\ 0 \end{pmatrix}.$$

Answer 2

For continuously differentiable $F$ defined on all of $\Bbb R^3$, the divergence of $F$ is zero exactly when $F$ is the curl of some other vector field $G$.