• Support PF! Buy your school textbooks, materials and every day products Here!

Helmholtz Decomposition

  • #1

Homework Statement



Let H(r) = x[tex]^{2}[/tex]yi + y[tex]^{2}[/tex]zj + z[tex]^{2}[/tex]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 = -[tex]\nabla[/tex][tex]\Psi[/tex] + [tex]\nabla[/tex]xA

So then:

F = -[tex]\nabla[/tex][tex]\Psi[/tex]

and G = [tex]\nabla[/tex]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)

[tex]\nabla[/tex][tex]^{2}[/tex][tex]\Psi[/tex] = - 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 [tex]\Psi[/tex] = -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?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
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.
 
Last edited:

Related Threads on Helmholtz Decomposition

Replies
1
Views
4K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
18
Views
3K
  • Last Post
Replies
2
Views
587
  • Last Post
Replies
1
Views
2K
Top