Helmholtz decomposition - motivation

Question

Our lecturer presented us the Helmholtz decomposition of smooth vector fields. He added a proof, but he didn't provide any single motivation - e.g. where Helmholtz used the decomposition or for which reasons is it mainly employed. Could anyone mention some motivation?

Answer 1

Physicist-to-be here!

Helmholtz's decomposition thm is useful when dealing with electromagnetism: with the notation of the wpedia page (http://en.wikipedia.org/wiki/Helmholtz_decomposition), when F is either the electric or magnetic field, you can employ Maxwell's Laws to find the formulas for the computing of electric/magnetic potentials.

Answer 2

Let's say you decomposed F = nabla X A + nabla.phi

Then,

nabla.F = nabla.nable.phi
nabla X F = nabla X nabla X A

If you integrate F, you only have to integrate nabla X A, because the nabla.phi part can be simplified using the chain rule into phi(endposition)-phi(startposition). integrating nabla X A can usually be simplified using stoke's theorem.