Green's Theorem Green's Theorem In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is the two-dimensional special case of the more general Stokes' theorem, and is named after British mathematician George Green. Let C be a positively oriented, piecewise smooth, simple closed curve in the plane 2, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, For positive orientation, an arrow pointing in the counterclockwise direction may be drawn in the small circle in the integral symbol. In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.
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The following is a proof of the theorem for the simplified area D, a type I region where C2 and C4 are vertical lines. A similar proof exists for when D is a type II region where C1 and C3 are straight lines. The general case can be deduced from this special case by approximating the domain D by a union of simple domains. his theorem is an application of the fundamental theorem of calculus to integrating a certain combinations of derivatives over a plane. It can be proven easily for rectangular and triangular regions. Both sides of its equality are finitely additive so that the result holds for any planar region practically all of which can be divided into triangles and rectangles. This proves the theorem for reasonably shaped regions. Its generalization to non-planar surfaces (proved directly from it by using the finite additivity of both sides) is Stokes' Theorem described below. Its formal statement is as follows: Let S be a sufficiently nice region in the plane, and let S be its boundary; then we have where the boundary, S is traversed counterclockwise on its outside cycle, (and clockwise on any internal cycles as you can verify using zippers.) Meaning of this theorem: Green's Theorem is a form that the fundamental theorem of calculus takes in the context of integrals over planar regions. Greenâ€™s theorem is a very important theorem of integration. We can also relate it with many theorem like strokes's theorem, gauss theorem. This theorem is mainly used for integration of a line combined with a curve plane. If we take line B and plane C, then, we can find the combination of that integration.
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Greenâ€™s theorem is used in to integrate derivatives in a particular plane. Here, we use basic theorem of integeration. By using greenâ€™s theorem, we will discuss some example problem. Here, we will leatn the proof of the green's theorem. Greeens theorem shows the relationship between a line integral and a surface integral. If a line integral is given, we can convert it to surface integral or double integral and viseversa using this theorem.
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Published on Jul 3, 2012
For positive orientation, an arrow pointing in the counterclockwise direction may be drawn in the small circle in the integral symbol. In ph...