- Let $F(x,y)$ be the vector field $2xyi−y2j$. What is the total outward flow of this field across the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$?
I have no idea how to do this problem. We just learned Green's theorem, so maybe something to do with that? I'm not sure honestly!
- Evaluate the integral of $\frac{y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$ where $c$ is the circle $x^2+y^2=1$.
For the second one, I know you can't use green's theorem becauase F isn't defined at the origin within the circle, so how would I do this?
Green's theorem says that $\oint M(x,y)dx+ N(x,y)dy= \int\int \left(\frac{\partial N}{\partial x}- \frac{\partial M}{\partial y}\right)dxdy$.
In the first problem $M= 2xy$ and $N= y^2$ (I assume your "y2" was intended to be $y^2$. You can write that as y^2) $\frac{\partial N}{\partial x}= 0$ and $\frac{\partial M}{\partial y}= 2x$ so the flow through that ellipse is $\int_{-a}^a\int_{-\frac{b}{a}\sqrt{a^2- b^2}}^{\frac{b}{a}\sqrt{a^2- x^2}} -2xdydx$.
For the second problem, I would change to polar coordinates. $x= r cos(\theta)$, $y= r sin(\theta)$, $dx= cos(\theta)dr- rsin(\theta)d\theta$, and $dy= sin(\theta)dr+ rcos(\theta)d\theta$.