A divergenceless vector field, also called a solenoidal field, is a vector field for which . Therefore, there exists a such that . Furthermore, can be written as

where

Following Lamb's 1932 treatise (Lamb 1993), and are called toroidal field and poloidal field.

More things to try:

Weisstein, Eric W. "Divergenceless Field." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DivergencelessField.html