Showing Greens First Theorem Integral is 0

[EngageEngage]

[SOLVED] Greens Functions

Homework Statement

show:
$$\int\int\int_{D}\vec{F}\cdot\vec{G}dV = 0$$
where:
$$\vec{F}=\nabla\phi$$
$$\vec{G}=\nabla\psi$$
$$\nabla\cdot\vec{F}=0$$
$$\psi|_{\partial D}=0$$

The Attempt at a Solution

This looks like a problem for Greens first theorem:

$$\int\int\int_{D}\phi\nabla^{2}\psi dV = \int\int_{\partial D}\phi\nabla\psi dS - \int\int\int_{D}\nabla\psi\cdot\nabla\phi dV$$

The very right term is clearly the integral that I'm looking for. So, i will set it to look like the requested answer. Also, I know that
$$\psi|_{\partial D}=0$$
meaning that I can also throw out the second term because that term wants me to integrate the gradient of psi over the surface, while I know that psi is 0 over the surface. So, I am left with this:

$$\int\int\int_{D}\phi\nabla^{2}\psi dV = - \int\int\int_{D}\vec{F}\cdot\vec{G} dV$$

So, this means that the term on the left mus equal zero. Does anyone know how I can show this? Psi is not zero through the domain, and the problem doesn't specify that it is a harmonic potential (although I suppose it could be). Could someone please help me with this step? Any help at all is greatly appreciated.

 

[Dick]

Uh, phi is given to be harmonic, not psi. Since the divergence of it's gradient is zero. Why don't you move the laplacian operator over to phi?

 

[Dick]

I.e. just switch the roles of psi and phi?

 

[EngageEngage]

Thanks for the help! This will also eliminate the middle term anyways too, since psi will be zero there on the boundary, right?

 

[Dick]

Thanks for the help! This will also eliminate the middle term anyways too, since psi will be zero there on the boundary, right?

Right.

 

[EngageEngage]

thanks a lot for the help Dick