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Line Integral

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The integral of a continuous function ALT= along a curve ALT= with continuously differentiable parametrization

$\displaystyle C: t\to p(t)\in \mathbb{R}^n\,,\,p'(t) \neq 0\,,\, t\in[a,b] \,, $

is defined by

$\displaystyle \int\limits_C f = \int\limits_a^b f(p(t))\vert p'(t)\vert\,dt $

and is independent of the parametrization. Particulary one gets the length of the curve by setting $ f=1$.

Weaker conditions on the smoothness of ALT= and $ p$ are possible by defining the integral as a suitable limit.

(Authors: App/Höllig/Höfert)

Examples:

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  automatically generated 8/ 4/2008