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Theorem of Gauß in the Plane

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The divergence theorem in the plane is a special case of the corresponding theorem of Gauß in space.

Let $ A$ be a region which is the interior of a closed curve $ C$ parametrized counterclockwise by $ \sigma: [a,b] \longrightarrow \mathbb{R}^2 .$ Let $ N(t)$ be the normal vector pointing outward. Then for each continuous differentiable vector field $ \Phi $

$\displaystyle \iint\limits_{A} \operatorname{div}\Phi \,dA = \int\limits_a^b \Phi (\sigma (t)) \cdot N(t) dt \ . $

Note that $ N(t) = (y'(t),-x'(t)) $ if $ \sigma (t) = (x(t),y(t)) .$

The integral

$\displaystyle \int\limits_{C} \Phi \cdot n dx := \int\limits_a^b \Phi (\sigma (t)) \cdot N(t) dt $

is called the flux of $ \Phi $ through $ C$ in the direction of the unit normal $ n = \frac{N(t}{\vert N(t)\vert} .$

Example:

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  automatically generated 7/ 4/2005