Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Recently, I had an exam and in that I was asked to evaluate the line integral of the function $$F=\frac{-y}{x^2+y^2}i+\frac{x}{x^2+y^2}j$$ alongside the unit circle, $0 \le t \le 2\pi $ . Moreover, it was asked if this integral could be carried out with using Green's Theorem or not and why?

For the first, I did the following: $$\oint_C F\cdot dr=\int_0^{2\pi}F\big (\cos(t),\sin(t)\big)\cdot\big(-\sin(t),\cos(t)\big)dt$$ which is $2\pi$. But always I have problem with above theorem and I do know I didn't pass this part of question correctly. May I ask to help me? Thank you.

share|cite|improve this question
up vote 2 down vote accepted

Your calculation of the line integral is correct. No, Green's Theorem cannot be applied directly because the vector field has a singularity at the origin.

Some years ago I taught a multivariable calculus course, and this very vector field was one of the stars of the whole show. You can see for instance $\S 5$ of these notes for an extended discussion of the fact that this vector field is irrotational -- i.e., has zero curl -- but is not conservative and in particular does not have a scalar potential function. Then, if you are feeling especially curious, you can go on to read the next section, which segues from this example to a discussion of De Rham cohomology and the problem that the main character in the film A Beautiful Mind writes on the blackboard in the multivariable calculus class attended by his future wife.

share|cite|improve this answer
@Oh Thanks Pete. I saw that film. Thanks for the refrences. I will follow what you noted. – Nancy Rutkowskie Aug 6 '12 at 14:00
I hope you enjoy the notes. Let me say that the one I linked to is "handout eight"; if you replace the "eight" by a different number you'll get other sections of the notes, and I believe that five through seven also have things to say about this vector field. (Or go to and scroll down to Multivariable Calculus to see the full list.) – Pete L. Clark Aug 6 '12 at 14:05
@PeteL.Clark, one question/aclaration: when you say "this vector field is irrotational yet not conservative...", you mean "in any domain containing the origin", otherwise $$\arctan\frac{y}{x}\,$$ is a potential. – DonAntonio Aug 6 '12 at 16:01
@DonAntonio: agreed. – Pete L. Clark Aug 6 '12 at 16:50

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.