16.4 Videos Guide

16.4a

Theorem (statement and partial proof):

• Green’s Theorem (essentially the Fundamental Theorem of Calculus for double integrals): For a region $D$ in $\mathbb{R}^{2}$ bounded by a positively oriented, simple, closed curve $C$ and a vector field $\mathbf{F}(x,\ y)=P(x,\ y)\mathbf{i}+Q(x,\ y)\mathbf{j}$, if $P$ and $Q$ have continuous partial derivatives on an open region containing $D$, then
  $$\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}=\oint^{\ }_{C}{P\ dx+Q\ dy}=\int^{\ }_{\partial D}{P\ dx+Q\ dy}=\iint^{\ }_{D}{\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\ dA}$$

16.4b

• Notation: $C=\partial D$ is the simple, positively oriented boundary curve of $D$. The symbol $\oint^{\ }_{C}\$ is used to indicate positive orientation.

Exercises:

• Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.

  • $\oint^{\ }_{C}{y\ dx-x\ dy}$, $C$ is the circle with center the origin and radius 4

16.4c

• $\oint^{\ }_{C}{x^{2}y^{2}\ dx+xy\ dy}$, $C$ consists of the arc of the parabola $y=x^{2}$ from $(0,\ 0)$ to (1, 1) and the line segments from $(1,\ 1)$ to $(0,\ 1)$ and from $(0,\ 1)$ to $(0,\ 0)$

16.4d

• Use Green’s Theorem to evaluate $\int^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}}$. (Check the orientation of the curve before applying the theorem.)
  $\mathbf{F}(x,\ y)=\left\langle e^{-x}+y^{2},\ {\ e}^{-y}+x^{2}\right\rangle$, $C$ consists of the arc of the curve $y=\cos x$ from $(-\frac{\pi}{2},\ 0)$ to $(\frac{\pi}{2},\ 0)$ and the line segment from $(\frac{\pi}{2},\ 0)$ to $(-\frac{\pi}{2},\ 0)$

• Green’s Theorem and regions with holes