Neumann Problem: Use the divergence theorem to show it has a solution

[ProtonHarvest]

From Partial Differential Equations: An Introduction, by Walter A. Strauss; Chapter 1.5, no.4 (b).

Homework Statement

"Consider the Neumann problem
(delta) u = f(x,y,z) in D

$$\frac{\partial u}{\partial n}=0$$ on bdy D."

"(b) Use the divergence theorem and the PDE to show that

$$\int\int\int_{D}f(x,y,z)dxdydz = 0$$

is a necessary condition for the Neumann problem to have a solution."

Homework Equations

Divergence Theorem:
$$\int\int\int_{D}\nabla\cdot \vec{f} d\vec{x} = \int\int_{S}f\cdot \vec{n}dS$$

In this case d$$\vec{x}$$ is a vector component I understand to translate to dxdydz, and I understand the rest of the divergence theorem.

The Attempt at a Solution

I wish I could but I don't know where to start; maybe I am rusty, maybe the question is worded strangely; maybe I'm over-thinking it. This is for a pure math PDE class, so even though I get why the condition is necessary from a physics perspective, I can't explain it in a math context, so I'm not sure how or what to apply the divergence theorem on.

If anyone can lend me a small clue to start me off that would be wonderful. I'm going to be checking this constantly so I will certainly respond.

 

[DarthBane]

Not to be critical, but check the number you posted. you have posted that you need to do ch 1.2 #4 (b). that is ch 1.5 #4 (b). So either typed it wrong or you are on the wrong problem, b/c I am in a course using the same book. If you are doing the correct problem, then I can say I have no earthly idea where to start.

 

[ProtonHarvest]

DarthBane, you're right, it is Chapter 1.5, I've edited it. Thanks for the heads up.