proof of Green’s theorem

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

\int\!\!\!\int_{R}\left(\frac{\partial g}{\partial x}-\frac{\partial f}{% \partial y}\right)\;dA=\int\!\!\!\int_{R}\frac{\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

\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

\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 $P=P_{1}+P_{2}$ , where $P_{1}=(A(y),y)$ and $P_{2}=(B(y),y)$ .

\int_{a}^{b}g(A(y),y)\;dy-\int_{a}^{b}g(B(y),y)\;dy=\int_{P_{1}}\vec{F_{1}}% \cdot 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

\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

\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

\int\!\!\!\int_{R}\left(\frac{\partial g}{\partial x}-\frac{\partial f}{% \partial y}\right)\;dA=\oint_{P}\vec{F_{1}}\cdot d\vec{t}+\oint_{P}\vec{F_{2}}% \cdot d\vec{t}=\oint_{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.

Title	proof of Green’s theorem
Canonical name	ProofOfGreensTheorem
Date of creation	2013-03-22 12:28:47
Last modified on	2013-03-22 12:28:47
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	11
Author	mathcam (2727)
Entry type	Proof
Classification	msc 26B12