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

I am working on the following exercise:

The function $f$ is called an integrating factor for the 1-form $\omega$ if $f({\bf x}) \neq 0$ for all $\bf x$ and $d(f\omega) = 0$. If the 1-form $\omega$ has an integrating factor, show that $\omega \wedge d\omega = 0$.

I am stuck here... I got $$d(f\omega) = df \wedge \omega + f \wedge d\omega = df \wedge \omega + f\ d\omega = 0$$ but that doesn't seem to get me anywhere. I also tried expanding this further (using the definition of $df$), but this gets quite ugly soon and didn't help either. The same goes for $\omega \wedge d\omega$.

share|cite|improve this question
Consider $\omega \wedge d\omega = \Omega$. Multiply it by $f$ ($f \neq 0$) and use your equation. – Nimza Feb 19 '12 at 19:39
up vote 2 down vote accepted

That last equation gives you $$d\omega = -(1/f) df\wedge\omega. $$ Then you have $$\omega \wedge d\omega = -(1/f)\omega\wedge df \wedge \omega = (1/f)\omega\wedge\omega \wedge df = 0.$$ This is because the wedge product of any form with itself is zero. Notice that it is critical that $f$ be zero-free, or the taking of the reciprocal breaks everything.

share|cite|improve this answer
Pay attention to the signi -. – Andrea Feb 19 '12 at 19:44
I think it should be $d\omega = -(1/f) df\wedge\omega$ (but that makes no difference). Thanks, working with differential forms I seem to forget that division exists. – koletenbert Feb 19 '12 at 20:06

If d(f w)=0 it implies that f w = d l (Pointcaré's lemma), where l is an scalar function. Therefore w= f^-1 d l. Taking the exterior derivative in both sides we get dw= - f^-2 df \wedge dl + 0 = - f^-1 df \wedge w.

And finally

w \wedge dw= - f^-1 w \wedge df \wedge w =0, qed.

share|cite|improve this answer
See this introductory note about posting mathematical expressions using $\LaTeX$ syntax. – hardmath Mar 31 at 17:27

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.