Section 6.1 Green’s Theorem

6.1 Green’s Theorem

In previous section, when we have a conservative vector field, we can use Fundamental Theorem of Line integral to do the line integration.

It is especially useful when the curve is a closed curve. We cannot apply the Fundamental Theorem of Line Integral if the vector field is not conservative. In this section, we introduce the Green’s Theorem when we do the line integral on a closed curve.

Theorem: Green’s Theorem

Let $D$ be an open, simply connected region with a boundary curve $C$ that is a piecewise smooth, simple closed curve oriented counterclockwise. Let $\mathbf{F}=<P,Q>$ be a vector field with component functions that have continuous partial derivatives on $D$. Then, 

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

Remark: Green’s Theorem also show that if $\mathbf{F}$ is conservative and we do line integral over a closed curve, the integration is $0$. 

Example 1: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $\mathbf{F}(x,y)=<x{y}^2,x^2>$ and $C$ is positively oriented along the rectangle with vertices $(0,0)$, $(5,0)$, $(5,4)$ and $(0,4)$. 

Exercise 1: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $\mathbf{F}(x,y)=<x{y}^2,y^2>$ and $C$ is positively oriented along the rectangle with vertices $(0,0)$, $(3,0)$, $(3,5)$ and $(0,5)$. 

Example 2: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $\mathbf{F}(x,y)=<e^y,2x{e}^y>$ and $C$ is positively oriented along the rectangle with vertices $(0,0)$, $(2,0)$, $(2,3)$ and $(0,3)$. 

Exercise 2: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $\mathbf{F}(x,y)=<x+\text{cos}(y),\text{sin}(x)>$ and $C$ is positively oriented along the rectangle with vertices $(0,0)$, $(\pi ,0)$, $(\pi ,\frac{\pi }{2})$ and $(0,\frac{\pi }{2})$. 

Example 3: Evaluate ${\int }_C(x-y)dx+(x^2+y)dy$ where $C$ is positively oriented along the triangle with vertices $(0,0)$, $(2,0)$, and $(2,3)$.

Exercise 3: Evaluate ${\int }_C(x+y)dx+(y-x^2)dy$ where $C$ is positively oriented along the triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$.

Example 4: Evaluate ${\int }_C(2y+\text{sin}(x))dx+(3x+e^{y^2})dy$ where $C$ is positively oriented along the boundary of the region enclosed by $y=x^2$ and $x=y^2$.

Exercise 4: Evaluate ${\int }_C(y+e^{x^3})dx+(2x+\text{cos}(y^4))dy$ where $C$ is positively oriented along the boundary of the region enclosed by $y=x^3$ and $x=y^3$.

Example 5: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $\mathbf{F}(x,y)=<\text{cos}(x)-y^2,\text{sin}(y)-x^2>$ and $C$ is the curve of $y=\text{sin}(x)$ from $(0,0)$ to $(\pi ,0)$ and then along the line segment from $(\pi ,0)$ to $(0,0)$.

Group work:

1. Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $\mathbf{F}(x,y)=<\text{ln}(x)+2y,\text{tan}(y)+x^3>$ and $C$ is the curve of $y=\text{cos}(x)$ from $(-\frac{\pi }{2},0)$ to $(\frac{\pi }{2},0)$ and then along the line segment from $(\frac{\pi }{2},0)$ to $(-\frac{\pi }{2},0)$ .

2. Find the work done by $\mathbf{F}(x,y)=<3+2x+y^2,2x^2+y>$ on a particle that moves from $(0,0)$ to $(0,1)$ then along the curve $y=(x-1{)}^2$ until $(1,0)$, and then move from the $(1,0)$ to $(0,0)$. 

3. Find the work done by $\mathbf{F}(x,y)=<y^3{e}^{xy},2y{e}^{xy}+x{y}^2{e}^{xy}>$ on a particle that moves along the curve $\overrightarrow{r(t)}=<\text{cos}(t),\text{sin}(t)>$, $0\le t\le 2\pi$.

4. Find the work done by $\mathbf{F}(x,y)=<2x+y,xy>$ on a particle that moves on a line segment from $P(0,1)$ to $Q(2,3)$.

5. Find the work done by $\mathbf{F}(x,y)=<y^2,2y+2x>$ on a particle that moves along the curve $\overrightarrow{r(t)}=<\text{cos}(t),\text{sin}(t)>$, $0\le t\le \frac{\pi }{2}$.