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Surface Integral

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The integral of a continuous function $ f$ over a surface $ S$ with a regular parametrization

$\displaystyle \left( \begin{array}{c} x_1 \\ \vdots \\ x_{n-1} \end{array}\righ...
...t( \begin{array}{c} y_1 \\ \vdots \\ y_{n} \end{array}\right)\,, \quad x \in R

and normal direction $ \xi$ is defined as

$\displaystyle \int\limits_S f \,dS=
\int\limits_R (f\circ s)\,\vert\det (\partial_1 s , \ldots , \partial_{n-1}, \xi)\vert\, dR

and is independent of the parametrization. The amount of the determinate is equal to the scaling factor of the elementary regions:

$\displaystyle dS = \vert\det (\partial_1 s , \ldots , \partial_{n-1}, \xi)\vert\, dR

In the special case $ f=1$ one gets the area of $ S$.

Weaker conditions on the smoothness of $ f$ and $ s$ are possible by defining the integral as a suitable limit. Also the surface can be composed of several smaller ones. In this case the integral over the region is the sum over the integrals of the smaller surfaces.

(Authors: Höllig/Much/Höfert)


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  automatically generated 5/30/2011