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Integration with Spherical Coordiantes

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview
For the coordinate transformation

$\displaystyle x = r\sin\vartheta\cos\varphi,\quad y = r\sin\vartheta\sin\varphi,\quad z = r\cos\vartheta $

we have

$\displaystyle dx\,dy\,dz = r^2 \sin\vartheta\,dr\,d\vartheta\,d\varphi \,. $

Particulary the integral of a function $ f$ on a sphere $ K: \; 0 \le r \le R$ is given by

$\displaystyle \int\limits_K f = \int\limits_0^{2 \pi}\int\limits_0^{\pi} \int\limits_0^{R} f(r,\vartheta,\varphi)r^2 \sin\vartheta\,dr\,d\vartheta\,d\varphi \,. $

(Authors: Höllig/Much/Höfert)

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  automatically generated 8/ 4/2008