Dimension of vector field in gauss divergence theorem

Question

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 $x,y,z\in\mathbb{R}$ ?

Answer 1

$F=xyz$ 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