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We've been learning about the divergence theorem and the Neumann problem.

I came across this question.

Use the divergence theorem and the partial differential equation to show that

[tex]\underbrace{\int\int\int}_{D}f(x,y,z)dxdydz=0[/tex] is a necessary condition for the Neumann problem to have a solution.

Where the Neumann problem is [tex]\Delta=f(x,y,z)[/tex] in D, [tex]\frac{\partial u}{\partial n}=0[/tex] on [tex]\partial D[/tex].