Helmholtz Decomposition - Longitudinal and Transverse Fields

Longitudinal and Transverse Fields

A terminology often used in physics refers to the curl-free component of a vector field as the longitudinal component and the divergence-free component as the transverse component. This terminology comes from the following construction: Compute the three-dimensional Fourier transform of the vector field F, which we call . Then decompose this field, at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k. So far, we have

Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier transforms, we derive:

so this is indeed the Helmholtz decomposition.