Helmholtz decomposition - motivation 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?
 A: 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.
A: 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.
A: One motivation would be to find different kind of wave motion in solids.
For an isotropic solid we have the following equations
$$\rho \frac{\partial^2 \mathbf{u}}{\partial t^2} = (\lambda + 2\mu) \nabla\nabla \cdot \mathbf{u} - \mu \nabla \times \nabla \times \mathbf{u} \enspace ,$$
using Helmholtz decompositions you can "translate" these equations into the following wave equations
$$\nabla^2 \phi = \frac{1}{\alpha^2} \frac{\partial^2 \phi}{\partial t^2}$$
and
$$\nabla^2 \mathbf{A} = \frac{1}{\beta^2} \frac{\partial^2 \mathbf{A}}{\partial t^2}\, ,$$
where $\alpha$ is the speed of longitudinal waves (P-waves) and $\beta$ is the speed of transverse wave (S-waves).
