closed but not exact

Question

I saw several times that $\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$ is closed but not exact. Closed, is obvious but I can't prove non exactness, can one please help me ?

My attempt, let $f\in \omega^{0}(U)$ and $df=\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y} dy$ and so $\frac{\partial f}{\partial x}(x,y)=\frac{-y}{x^2+y^2},\frac{\partial f}{\partial y}(x,y)=\frac{x}{x^2+y^2}$. Now let $g(\theta)=f(\cos \theta,\sin \theta)$ so $g'(\theta)=1 \implies g(\theta)=\theta+C$. Now i'm not getting any contradiction.

Well, the final step is missing, that $\theta$ is not well-defined on all of $\mathbb{R}^2\setminus {(0,0)}$, so $g$ is only locally definable, not globally. [Note that every closed form is locally exact.] — Daniel Fischer, Oct 21 '14 at 15:56
Nothing wrong with your attempt, until the final sentence. You do have a contradiction, because $\theta$ is not well defined single valued continuous function. — Harald Hanche-Olsen, Oct 21 '14 at 15:56
Ok so, can i also say like this ? $f$ is $C^{\infty}$ so $\frac{\partial f}{\partial x}$ should be continuous but its not at ${0,0}$ — dragoboy, Oct 21 '14 at 16:03
the form you had taken is actually a closed form. So it wouldn't help the way you are trying. — permutation_matrix, Dec 14 '22 at 17:00

mookid · Answer 1 · 2014-10-22T17:32:50.260

1

You know that if you integrate over a loop into which there is no singularity, as the form is exact the integral is 0.

Then let us try with a loop containing 0 inside: let us take the circle $C(0,1)$.

$$\int_{C(0,1)} -\frac y{x^2 + y^2}dx + \frac x{x^2 + y^2}dy = \int_0^{2\pi} -\frac {r\sin\theta}{r^2} (-r \sin\theta d\theta) + {r\cos\theta}{r^2} (r \cos\theta d\theta)\\ = 2\pi \neq 0 $$

edited Oct 22 '14 at 17:32

answered Oct 21 '14 at 16:46

mookid

28,236

Ok i got it, but I've not seen yet proof of the fact that exact iff that integration over a closed curve is zero. Can you please provide a reference for that ? – dragoboy Oct 21 '14 at 17:00
then what is your definition of exact? – mookid Oct 21 '14 at 17:46

closed but not exact

1 Answers1