Divergenceless Field

A divergenceless field, also called a solenoidal field, is a vector field for which . A sufficiently smooth divergenceless field can locally be written as , where is a vector potential.

Fix an origin, let be the position vector from it, and write . On a spherical shell centered at this origin, a sufficiently smooth divergenceless field satisfying for each sphere in the spherical shell can be written as

			(1)
			(2)

where and are scalar functions and

			(3)
			(4)
			(5)
			(6)

The components and are called the toroidal field and poloidal field, respectively (Backus 1986). In general, these components change when the chosen origin changes. Here is the gradient, is the curl, and is the Laplacian.