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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:```$$ \int_E \mathrm{div} f(x)\, dx
 = \int_{\partial^* E} \langle \nu_E(x),f(x)\rangle \,d\mathcal H^{n-1}(x)$$```

See http://planetmath.org/?method=l2h&from=objects&name=GaussGreenTheorem&op=getobj for the details.

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 is a sufficient condition 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

2 added 381 characters in body; added 25 characters in body

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:```$$ \int_E \mathrm{div} f(x)\, dx
 = \int_{\partial^* E} \langle \nu_E(x),f(x)\rangle \,d\mathcal H^{n-1}(x)$$```

See http://planetmath.org/?method=l2h&from=objects&name=GaussGreenTheorem&op=getobj for the details.

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 is a sufficient condition 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

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# 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:```$$ \int_E \mathrm{div} f(x)\, dx
 = \int_{\partial^* E} \langle \nu_E(x),f(x)\rangle \,d\mathcal H^{n-1}(x)$$```

See http://planetmath.org/?method=l2h&from=objects&name=GaussGreenTheorem&op=getobj for the details.

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?
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