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Greens Theorem

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Green's theorem may be considered as a special case of the integral theorem of Stokes in space.

Standard Version.

Let $ p(x,y)$ and $ q(x,y)$ be continuous differentiable functions on a region $ A$ which is the interior of a closed path $ C$ . Suppose that $ C$ is parametrized counterclockwise and let $ \Phi (x,y) = (p(x,y), q(x,y)$ Then

$\displaystyle \int_C \Phi dx = \iint_A q_x - p_y dy dx \ .$

General Version.

Let $ A$ be a region in the plane whose boundary consists of a finite number of smooth curves $ C_1, \ldots , C_m$ . Assume that each curve $ C_i$ of the is parametrized by $ \sigma_i :[a_i,b_i] \longrightarrow \mathbb{R}^2$ in such a way that $ A$ lies to the left of the curve. Let $ \Phi (x,y) = (p(x,y), q(x,y)) $ be a continuous differentiable vector field.

Then

$\displaystyle \iint_A \operatorname{curl} \Phi \, dA \ = \ \iint_A q_x - p_y dy... ...hi(\sigma_m (t)) \cdot \sigma_m' (t) \ = \ \int\limits_{C} \Phi \cdot dx \ . $

Note that in the general version the boundary curves need not form a closed path.

Green's theorem expresses the curve integral of a vector field as a double integral over its scalar curl. Sometimes the calculation of the double integral is easier than that one of the curve integral and vice versa.

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  automatically generated 6/ 2/2008