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Gauss Green theorem
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(Theorem)
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Theorem 1 (Gauss-Green) Let $\Omega\subset \R^n$ be a bounded open set with $C^1$ boundary, let $\nu_\Omega\colon \partial \Omega\to \R^n$ be the exterior unit normal vector to $\Omega$ in the point $x$ and let $f\colon \overline{\Omega}\to \R^n$ be a vector function in $C^0(\overline\Omega,\R^n)\cap C^1(\Omega,\R^n)$ . Then $$ \int_\Omega \mathrm{div} f(x)\, dx =\int_{\partial \Omega} \langle f(x),\nu_\Omega(x)\rangle \, d\sigma(x). $$
Some remarks on notation. The operator $\mathrm{div} f$ is the divergence of the vector field $f$ , which is sometimes written as $\nabla \cdot f$ . In the right-hand side we have a surface integral, $d\sigma$ is the corresponding area measure on $\partial \Omega$ . The scalar product in the second integral is sometimes written as $f_n(x)$ and represents the normal component of $f$ with respect to $\partial \Omega$ ; hence the whole integral represents the flux of the vector field
$f$ through $\partial \Omega$ ;
This theorem can be easily extended to piecewise regular domains. However the more general statement of this Theorem involves the theory of perimeters and $BV$ functions.
Theorem 2 (generalized Gauss-Green) Let $E\subset \R^n$ be any measurable set. Then $$ \int_E \mathrm{div} f(x)\, dx = \int_{\partial^* E} \langle \nu_E(x),f(x)\rangle \,d\mathcal H^{n-1}(x) $$ holds for every continuously differentiable function $f\colon \R^n\to\R^n$ with compact support (i.e. $f\in\mathcal C^1_c(\R^n,\R^n)$ ) where
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"Gauss Green theorem" is owned by paolini.
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Cross-references: Hausdorff measure, normal vector, subset, essential boundary, support, compact, continuously differentiable, measurable set, perimeters, theory, domains, regular, piecewise, theorem, flux, component, normal, represents, scalar product, area measure, integral, surface, side, vector field, divergence, operator, function, point, vector, unit normal, exterior, boundary, open set, bounded
There are 5 references to this entry.
This is version 10 of Gauss Green theorem, born on 2005-02-11, modified 2005-02-18.
Object id is 6741, canonical name is GaussGreenTheorem.
Accessed 10029 times total.
Classification:
| AMS MSC: | 26B20 (Real functions :: Functions of several variables :: Integral formulas ) |
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Pending Errata and Addenda
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