I want to find a closed 1form $w$ on $$M=\{(x,y,z):x^2y^2z^2=1\}\subset \mathbb R^3$$ which is not exact. I think that $$\frac{x\,\mathrm{d}x+y\,\mathrm{d}y+z\,\mathrm{d}z}{(x^2+y^2+z^2)^{3/2}}$$ is a such one, but how can I prove that?

Have you even tried to compute the differential of the form to check that it vanishes, for example? Also, you think that this form works... Why? Does this problem come from a book, an assignment you were given...? In general it's a good idea to include such information. – Najib Idrissi Jul 10 '15 at 11:38

Your $1$form, regarded as a $1$form on $\Bbb R^3$ is the exterior derivative of $(x^2 + y^2 + z^2)^{1 / 2}$, hence it is exact, and hence so is its pullback to $M$. – Travis Willse Jul 10 '15 at 11:39
3 Answers
First, a little bit of algebraictopological intuition. Your set $M$ is a hyperboloid of one sheet, which means it deformation retracts to a circle. A closed, nonexact oneform is a nontrivial cohomology class: It should tell you that a loop around the "waist" of the hyperboloid is not contractible.
So look at the circle. Do you know a closed, nonexact oneform on $S^1 = \{x^2 + y^2 = 1\ \ (x,y)\in\Bbb{R}^2\}$? Can you adapt that inspiration to construct a closed, nonexact oneform on $M$?
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Hint A standard example of a $1$form that is closed but not exact is the form $$\frac{z \,dy + y \,dz}{y^2 + z^2}$$ on the punctured plane $\Bbb R^2  \{0\}$ (with standard coordinates suggestively named); suggestively (but not quite precisely) this is often denoted $d \theta$, where $\theta$ is an angular polar coordinate.
Remark Incidentally, if one raises the sole index of this form using the Euclidean metric, the result is the vortex vector field, $\frac{1}{y^2 + z^2}(z \partial_y + y \partial_z)$.
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@Neal You're right, of course (I wrote that after typesetting the last remark about vector fields). – Travis Willse Jul 10 '15 at 11:52
This one is exact, unfortunately: it is just $$\mathrm d\biggl(\frac{1}{(x^2+y^2+z^2)^{1/2}}\biggr).$$
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