The main purpose of Green’s Theorem is that it helps us find line integrals which will otherwise be difficult to integrate, in particularly piecewise-smooth curves where they are defined using multiple position vectors.
We shall look at one such example.
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USES OF GREEN'S THEOREM
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VIDEO LECTURE
MAIN CONCEPTS
Given certain conditions, Green's theorem allows us to evaluate close loop line integrals easily which would otherwise be difficult, in particularly when dealing with piecewise-smooth curves. Get the Printer Friendly Version COMMENTS
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CONCEPT
A particle moves in a counter clockwise direction around a rectangle having the vertices (0, 0), (6, 0), (0, 4), and (6, 4) under the influence of the vector field
Our objective is to find the work done by
The work done is given by the close loop line integral
Noticing that the curve is piecewise-smooth, if we were to find this integral directly, we need to express the position vector An easier method is to use Green’s theorem. Pay close attention that all the conditions are satisfied: C is simply closed positively oriented and piecewise-smooth AND the vector field has continuous first partial derivatives throughout D. So by Green’s theorem, we can write,
using the appropriate rules of evaluating double integrals. We learn here that given certain conditions, close loop line integrals can easily be calculated using Green’s theorem. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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after one complete circuit.
into 4 parts, namely position vectors describing the edges of the curve.
