Green's Theorem: Difference between revisions

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imported>Emil Gustafsson
(New page: Green's Theorem is vector identity that is equivalent whith the curl theorem in two dimensions. It relates the line integral around a simple closed curve <math>\partial\Omega</math> an...)
 
imported>Emil Gustafsson
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Green's Theorem is vector identity that is equivalent whith the [[curl theorem]] in two dimensions. It relates the line integral around a simple closed curve <math>\partial\Omega</math> and the doubble integral over the plane region <math>\Omega\,</math>.
Green's Theorem is a vector identity that is equivalent to the [[curl theorem]] in two dimensions. It relates the line integral around a simple closed curve <math>\partial\Omega</math> with the doubble integral over the plane region <math>\Omega\,</math>.


The theorem is named after the british mathematician [[George Green]]. It can be applied to variuos fields in physics, among others flow integrals.
The theorem is named after the british mathematician [[George Green]]. It can be applied to variuos fields in physics, among others flow integrals.

Revision as of 01:19, 15 July 2008

Green's Theorem is a vector identity that is equivalent to the curl theorem in two dimensions. It relates the line integral around a simple closed curve with the doubble integral over the plane region .

The theorem is named after the british mathematician George Green. It can be applied to variuos fields in physics, among others flow integrals.

Mathematical Statement

Let be a region in with a positively oriented, piecewise smooth, simple closed boundary . and are functions defined on a open region containing and have continuous partial derivatives in that region. Then Green's Theorem states that

The theorem is equivalent to the curl theorem in the plane and can be written in a more compact form as