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Stokes Theorem

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Let $ S$ be a bounded orientable surface parametrized by $ \sigma (t,u)$ with normal vector $ N(t,u) = \sigma_t \times \sigma_u .$ Define the top side of $ S$ as this side where $ N(t,u)$ points outward. Assume that the boundary $ \partial S$ is parametrized in such a way that the top side of $ S$ lies on the left and assume that $ \partial S$ consists of finitely many smooth curves.

Then for each continuous differentiable vector field $ \Phi $ defined on an open set containing $ S$ and its boundary

$\displaystyle \iint\limits_S \operatorname{curl} \Phi \cdot \frac{N(t,u)}{\vert N(t,u)\vert} d\sigma = \int\limits_{\partial S} \Phi dx . $

The theorem of Stokes expresses the flux of the curl of $ \Phi $ through $ S$ as the curve integral of $ \Phi $ along the boundary of $ S .$ The special case when $ S$ is contained in the $ x,y$ - plane is known as Green's theorem.

The theorem does not apply to non-orientable surfaces. A Moebius strip is an example for a non-orientable surface in \mathbb{R}^3.

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  automatically generated 7/ 4/2005