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Green's Theorem

Green’s theorem was first introduced by a famous mathematician George Green. It is considered as a very important theorem in the field of mathematics and vectors as well as in physics and engineering. It is related to Stoke's Theorem in many ways. Green's theorem is said to be a two-dimensional special case of famous Stokes' theorem.

Green's theorem is the relation between a line integral about a closed curve and a double integral in a two-dimensional region which is bounded by the given closed curve.

Let us consider that C is a closed smooth curve placed in a two-dimensional space $R^{2}$. The curve is said to be positively oriented. The curve C is traced in an anticlockwise direction. Let us suppose that D be the interior of C. It is a region that is bounded by curve C. Region D has a smooth surface and there are no holes in it. The curve and the region D are sketched below -
Closed Curve Region
Suppose that P and Q are two functions of (x, y). These are defined in region D. P and Q are differentiable functions and possess continuous partial derivatives.

Green's theorem states the following formula -

$\oint_{C}(P\ dx+Q\ dy)=\iint_{D}(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dx\ dy$
Green's theorem is widely used in determining the area of the planes and their centroids. It is also used very often in physics and engineering to calculate flow integrals and in many other applications.

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