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In Gauss Divergence Theorem , $$\int \int_{\partial V} \textbf{F}\cdot\textbf{n}\,dS =\int \int \int_V \nabla\cdot\textbf{F}\,dx\,dy\,dz,$$ is there any restriction on the vector field $\textbf{F}$ ? Does it need to be in dimension three only ? Can I let it be $\textbf{F}=xyz$ where $x,y,z\in\mathbb{R}$ ?

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The concepts in the Divergence Theorem and Stoke's Theorem generalize to higher dimensions. Such generalizations make use of differential forms. See en.wikipedia.org/wiki/Stokes%27_Theorem#General_formulation –  Ron Gordon Dec 28 '12 at 18:12

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$F=xyz$ is not a vector field. You can take a look at the wikipedia definitions. Yes, $F$ has to be continuous and differentiable.

See here: http://en.wikipedia.org/wiki/Divergence_theorem

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