Greens Theorem

Applies to Line Integrals

Theorem §

Let R be a simply connected region with piecewise smooth curves C, oriented counterclockwise.

$$\oint Mdx+Ndy=\iint \left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dA,M=fx,N=fy$$

Examples §

Example 1 §

Use Green’s Theorem to find the line integral

$${\oint }_C{x}^{3}dx+xydy$$

for C is $(0,0)\to (1,0)\to (1,1)$

Solution §

$$\begin{array}{rl}S& ={\int }_{x=0}^{x=1}{\int }_{y=0}^{y=x}(y-0)dydx\\ & =\frac{x^3}{6}{|}_0^1\\ & =\frac{1}{6}\end{array}$$

Example 2 §

Use Green’s theorem to find line integral of

$$\oint x^{2}y+x^{3}dx+2xydy$$

where path is the region bounded by $y=x$, $y=x^3$

Solution §

$$\begin{array}{rl}S& ={\int }_{x=0}^{x=1}{\int }_{y=x^3}^{y=x}(2y-x^2)dydx\\ & ={\int }_0^1{x}^2-x^6-x^3+x^{5}dx\\ & =0.1074\end{array}$$

Example 3 §

Use Green’s theortem to find work done by force $\vec{F}$ on particle moving counterclockwise around the closed path $C:r=2\cos \theta$

$$\vec{F}=(e^x-3y)\hat{i}+(e^y+6x)\hat{j}$$

Solution §

$$\begin{array}{rl}W& =\oint \vec{F}dr\\ & ={\iint }_R{f}_{yx}-f_{xy}dA\\ & ={\int }_{\theta =0}^{\theta =\pi }{\int }_{r=0}^{r=2\cos \theta }(6-(-3))rdrd\theta \\ & =9\pi \end{array}$$