# Divergence theorem on non compact sets of R3

Lebesgue

## Main Question or Discussion Point

So my question here is: the divergence theorem literally states that
Let $\Omega$ be a compact subset of $\mathbb{R}^3$ with a piecewise smooth boundary surface $S$. Let $\vec{F}: D \mapsto \mathbb{R}^3$ a continously differentiable vector field defined on a neighborhood D of $\Omega$.
Then:
$\int_{\Omega} \nabla \cdot \vec{F} dxdydz = \oint_S \vec{F} \cdot \vec{n} dS$

My problem here is: why people (and with which argument) use this divergence or Gauss theorem to compute the electric field of some NOT bound set (for example, the typical infinte cylinder) of surface charge.

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Orodruin
Staff Emeritus