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Question

# Let $\mathbf{F}$ be a constant vector field. Show that$\iint_{\lambda S} \mathbf{F} \cdot \mathbf{n} d S=0$for any "nice" solid S. What should we mean by "nice"?

Solution

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Since ${\bf F}=(a,b,c)$ is a constant vector field, then:

$\nabla\cdot{\bf F}=\dfrac{\partial (a)}{\partial x}+\dfrac{\partial (b)}{\partial y}+\dfrac{\partial (c)}{\partial z}=0$

Then, using the Gauss's Divergence Theorem we get:

$\iint_{\delta S}{\bf F}\cdot{\bf n}dS=\iiint_V\nabla\cdot{\bf F}dV=0$

A "nice" solid is a solid enclosed by a smooth and orientable closed surface.

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