# Generalized Gauss-Green theorem

I am looking for a generalized version of the Gauss-Green theorem also known as the divergence theorem:

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

A quick search on MathSciNet suggests that there are generalizations for bad domains and nonsmooth functions. However, they seem to rely on heavy machinery and not to be suited for the special case I am interested in.

For example, I found this formula on PlanetMath:$$\int_E \mathrm{div} f(x)\, dx = \int_{\partial^* E} \langle \nu_E(x),f(x)\rangle \,d\mathcal H^{n-1}(x)$$

Let $\Omega \subset \mathbb{R}^n$ be open and bounded and $f\in C^1(\Omega, \mathbb{R}^n) \cap C^0(\overline\Omega, \mathbb{R}^n)$.

Question: What conditions do we have to impose on $\Omega$ (or $f$) to ensure that the divergence theorem holds true?

To clarify my question: I know that requiring the boundary of $\Omega$ to be piecewise regular is sufficient for the Gauss-Green theorem to be true. I wondered if this condition is also necessary. If so: is the an other "version" of Gauss-Green (e.g. the one cited above) which holds true under weaker conditions and is especially suited for the case of an open and bounded domain

• Rather than a generalization of Gauss-Green theorem, the divergence theorem is the $3$-dimensional version of Stokes theorem, of which the Gauss-Green theorem itself is the $2$-dimensional version. Jun 7, 2011 at 21:51
• Wikipedia says that the divergence theorem is also known as Gauss' theorem, Ostrogradsky's theorem, or Gauss–Ostrogradsky theorem. My professor called it the Gauss-Green theorem. Jun 7, 2011 at 22:09
• You are yet to say precisely what conclusions you want to hold. Exactly what theorem do you want to be true? And do you have reason to doubt the first statement on the PlanetMath page? Jun 7, 2011 at 22:11