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## Homework Statement

Let

**H**(

**r**) = x[tex]^{2}[/tex]y

**i**+ y[tex]^{2}[/tex]z

**j**+ z[tex]^{2}[/tex]x

**k**. 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]x

**A**

So then:

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

and

**G**= [tex]\nabla[/tex]x

**A**

## 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?