solenoidal field

A solenoidal vector field is one that satisfies

$$\nabla \cdot 𝐁=0$$

at every point where the vector field $𝐁$ is defined. Here $\nabla \cdot 𝐁$ is the divergence.

This condition actually implies that there exists a vector $𝐀$, such that

$$𝐁=\nabla \times 𝐀.$$

For a function $f$ satisfying Laplace’s equation

$${\nabla }^{2}f=0,$$

it follows that $\nabla f$ is solenoidal.