Question

Computing the integral of a vector over closed path?
can someone help me with this problem
let v=(-y/x^2+y^2 , x/x^2+y^2)
compute the integral of v over a cirlcle C(0,R)
compute integral of v over a closed path gamma

Answer 1

You need the parametrization of the circle so that you can get \(\mathbb{v}(\mathbb{r}(t))\cdot \mathbb{r}'(t)\).

Answer 2

and what about the path gamma ?

Answer 3

can you post the solution please

Answer 4

What's the radius of the circle?

Answer 5

(0,r) that's what the problem says

Answer 6

Hmm... let's assume that the radius of the circle is some number \(a\) and the center of the circle is (0,R).
The circle can be parametrized by letting \(\mathbf{r}(t)=(a\cos(t)+0)~\mathbf{i}+(a\sin(t)+R)~\mathbf{j}, \; 0\leq t\leq2\pi\) so that \(\mathbf{r}'(t)=-a\sin(t)~\mathbf{i}+a\cos(t)~\mathbf{j}\).
Now, since \[\mathbf{v}(x,y)=\frac{-y\mathbf{i}+x\mathbf{j}}{x^2+y^2}\]
can you find \(\mathbf{v}(\mathbf{r}(t))\)?

Answer 7

I really don't know how to, do you mind if you showed me please kind stranger :)

Answer 8

Just substitute the values of x(t) and y(t) from r into v.

Answer 9

ok thanks , and for the path gamma what should i do ?

Answer 10

Oopsies. I called it r instead of \(\gamma\). Just pretend the r's are gamma.

Answer 11

alright thanks