Question

Is function u(r,\Theta) = r^2*sin(2 \Theta)) harmonic?

Answer 1

What is the definition of harmonic? I know, but I'm asking you to get you to think through this.

Answer 2

\[\Delta u=0\] I know the theory but polar coordinates is confusing me.

Answer 3

Ah, good. So now what is the Laplacian operator in polar coordinates?

Answer 4

It's this

Answer 5

So do the differentiation and see if it is zero.

Answer 6

When i derive in respect of r, should I use the product rule for derivatives, or is it just 2r*sin(2theta)?

Answer 7

It's a partial derivative, so just the second thing you wrote down.

Answer 8

so the answer is clearly no. Thanks :)

Answer 9

Exactly.