Section 6.2 Extended Versions of Green’s Theorem

6.2 Extended Versions of Green’s Theorem

Green’s Theorem can be extended to apply to regions with holes, that is, regions that are not simply-connected.

Theorem: Green’s Theorem

Let $D$ be a region with finitely many holes (so that $D$ has finitely many boundary curves), and denote the boundary of $D$ by $\partial D$. $\partial D$ is positive oriented. Let $\mathbf{F}=<P,Q>$ be a vector field with component functions that have continuous partial derivatives on $D$. Then, 

$${\int }_C\mathbf{F}\cdot d\overrightarrow{r}={\int }_{C}Pdx+Qdy=\int {\int }_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA.$$

Example 1: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $\mathbf{F}(x,y)=<x^2-y^2,x^2+\text{sin}(y^2)>$ and $C$ is the boundary of the region between the circle $x^2+y^2=1$ and $x^2+y^2=4$ with positive orientation. 

Exercise 1: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $\mathbf{F}(x,y)=<y^2,x+e^{y^2}>$ and $C$ is the boundary of the region between the circle $x^2+y^2=4$ and $x^2+y^2=9$ with positive orientation. 

Example 2: Evaluate ${\int }_C{y}^{3}dx-x^{3}dy$, where $C$ includes the two circles of radius $2$ and radius $1$ centered at the origin, both with positive orientation.

Exercise 2: Evaluate ${\int }_{C}x{y}^{3}dx+y{x}^{3}dy$, where $C$ includes the two circles of radius $3$ and radius $2$ centered at the origin, both with positive orientation.

Example 3: Evaluate ${\int }_{C}y{x}^{2}dx+x^{3}dy$ where $C$ is positively oriented along the ellipse $4x^2+y^2=4$. 

Exercise 3: Evaluate ${\int }_C(xy)dx+(x^2)dy$ where $C$ is positively oriented along along the ellipse $x^2+9y^2=9$. 

Example 4: Find the area between ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and circle $x^2+y^2=25$.

Exercise 4: Find the area between ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and circle $x^2+y^2=4$.

Example 5: Find area using planimeter and $\mathbf{F}(x,y)=<\frac{-1}{2}y,\frac{1}{2}x>$. 

Planimeter for map measure.

Group work:

1. Find the area of the region bounded by hypocycloid $\overrightarrow{r(t)}=<{\text{cos}}^3(t),{\text{sin}}^3(t)>$. The curve is parameterized by $t\in [0,2\pi ]$.

2. Find the counterclockwise circulation of field $\mathbf{F}(x,y)=xy\overrightarrow{i}+y^2\overrightarrow{j}$ around and over the boundary of the region enclosed by curves $y=x^2$ and $y=x$ in the first quadrant and oriented in the counterclockwise direction.

3. Find the work done by $\mathbf{F}(x,y)=<2y+2x\text{sin}(y),x^2\text{cos}(y)-3y^2>$ on a particle that moves along triangle $C$ with vertices $(0,0),(1,0)$ and $(1,1),$ oriented counterclockwise.

4. Evaluate the line integral ${\int }_{C}x{e}^{-2x}dx+(x^4+2x^2{y}^2)dy$ where $C$ is the boundary of the region between circles $x^2+y^2=4$ and $x^2+y^2=9$, and is a positively oriented curve.