What is the Helmholtz Decomposition of a Vector Field?

[Zebrostrich]

Homework Statement

Let H(r) = x$$^{2}$$yi + y$$^{2}$$zj + z$$^{2}$$xk. Find an irrotational function F(r) and a solenoidal function G(r) such that H(r) = F(r) + G(r)

Homework Equations

From Helmholtz's theorem, any vector field H can be expressed as:

H = -$$\nabla$$$$\Psi$$ + $$\nabla$$xA

So then:

F = -$$\nabla$$$$\Psi$$

and G = $$\nabla$$xA

The Attempt at a Solution

Taking the divergence of H(r) = F(r) + G(r), I obtained (since the Divergence of G is zero)

$$\nabla$$$$^{2}$$$$\Psi$$ = - 2xy - 2yz - 2zx

I really have no idea how to solve this equation. If I took the curl, I would have an even more complicated system. I found out a solution to this equation, but merely by guessing. That would be $$\Psi$$ = -xyz(x+y+z), and from there I found the two vector fields. However, that does not seem sufficient enough. Is there a better way to approach this problem that I am missing?

 

[Dick]

I don't think so. If you can guess a solution to Laplace's equation, which you did, you are way ahead of the game. I think that's the way you were intended to solve it. The problem was rigged that way. Great job.