PlanetMath: proof of Green's theorem

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proof of Green's theorem

(Proof)

Consider the region bounded by the closed curve in a simply connected space. can be given by a vector valued function $\vec{F}(x,y)=( f(x,y), g(x,y))$ . The region can then be described by

$\displaystyle \int\!\!\!\int_R \left(\frac{\partial g}{\partial x} - \frac{\par... ...partial g}{\partial x}\;dA - \int\!\!\!\int_R \frac{\partial f}{\partial y}\;dA$

The double integrals above can be evaluated separately. Let's look at

$\displaystyle \int\!\!\!\int_R \frac{\partial g}{\partial x}\;dA = \int_a^b\int_{A(y)}^{B(y)}\frac{\partial g}{\partial x}\;dxdy$

Evaluating the above double integral, we get

$\displaystyle \int_a^b (g(A(y),y) - g(B(y),y))\;dy = \int_a^b g(A(y),y)\;dy - \int_a^b g(B(y),y)\;dy$

According to the fundamental theorem of line integrals, the above equation is actually equivalent to the evaluation of the line integral of the function $\vec{F}_1(x,y)=( 0, g(x,y))$ over a path

, where

and

$\displaystyle \int_a^b g(A(y), y)\;dy - \int_a^b g(B(y), y)\;dy = \int_{P_1} \v... ... d\vec{t} + \int_{P_2}\vec{F_1}\cdot d\vec{t} = \oint_P \vec{F_1}\cdot d\vec{t}$

Thus we have

$\displaystyle \int\!\!\!\int_R \frac{\partial g}{\partial x}\;dA = \oint_P \vec{F_1}\cdot d\vec{t}$

By a similar argument, we can show that

$\displaystyle \int\!\!\!\int_R \frac{\partial f}{\partial y}\;dA = -\oint_P \vec{F_2}\cdot d\vec{t}$

where $\vec{F}_2=( f(x,y), 0)$ . Putting all of the above together, we can see that

$\displaystyle \int\!\!\!\int_R \left(\frac{\partial g}{\partial x} - \frac{\par... ..._P (\vec{F}_1 + \vec{F}_2)\cdot d\vec{t}=\oint_P (f(x,y), g(x,y))\cdot d\vec{t}$

which is Green's theorem.

"proof of Green's theorem" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Cross-references: Green's theorem, argument, similar, path, equivalent, equation, line integrals, integrals, function, vector, simply connected, closed curve, bounded, region
There is 1 reference to this entry.

This is version 8 of proof of Green's theorem, born on 2002-02-25, modified 2004-09-07.
Object id is 2690, canonical name is ProofOfGreensTheorem.
Accessed 12643 times total.

Classification:

AMS MSC:

26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

Pending Errata and Addenda

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Discussion

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Fundamental theorem of line integrals by castilladelcarpio on 2007-03-21 06:33:58

Mathcam, I am beginning my study of line integrals (I did study double integrals some months ago already). Could you please tell me which theorem you refer to when you say "the FT of Line integrals" and how do you apply it ot the proof? Thanks and excuse the bad english.

[ reply | up ]

Proof of green's theorem by logamath on 2007-02-03 02:23:47

The post by mathcam states that the region "R is described" and states the equality to the double integral. However, the double integral relates the difference of two 'volumes' to the sum of two areas (countour integration). The region R is therefore more accurately described by stating that the equality involves a relationship between areas and volumes. I think this would make Green's theorem more understandable.

Unsoundness of green's theorem: The equality to the double integral is not always true even under the required conditions, i.e. closed curve, continuous partial derivatives. Comments?

[ reply | up ]

Re: Proof of green's theorem by castilladelcarpio on 2008-11-17 09:45:45

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