Can Helmholtz Decomposition Simplify Complex Vector Fields?

[Zebrostrich]

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?

 

[mark726]

Thank you for your post. Your approach to this problem is correct, but there are a few things that can be improved upon.

Firstly, when taking the divergence of H, you should use the product rule for differentiation. This will give you the correct equation:

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

From here, you can use the method of separation of variables to solve for \Psi. This involves assuming a solution of the form \Psi(x,y,z) = X(x)Y(y)Z(z), and then plugging it into the equation and solving for each of the functions X, Y, and Z separately.

Once you have found \Psi, you can then find F and G using the equations you mentioned: F = -\nabla\Psi and G = \nablaxA. Again, you can use the product rule for differentiation to find the components of A.

Overall, your approach is correct, but it would be good to include some more details and steps in your solution. This will help make your solution more clear and understandable.

I hope this helps. If you have any further questions, please feel free to ask.