From Partial Differential Equations: An Introduction, by Walter A. Strauss; Chapter 1.5, no.4 (b).(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

"Consider the Neumann problem

(delta) u = f(x,y,z) in D

[tex]\frac{\partial u}{\partial n}=0[/tex] on bdy D."

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

[tex]\int\int\int_{D}f(x,y,z)dxdydz = 0[/tex]

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

2. Relevant equations

Divergence Theorem:

[tex]\int\int\int_{D}\nabla\cdot \vec{f} d\vec{x} = \int\int_{S}f\cdot \vec{n}dS [/tex]

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

3. 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.

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

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