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I wonder how to check some simple 1-form or 2-form is exact. For instance, 2-form $w=xdy \wedge dz + ydz \wedge dx − 2zdx \wedge dy$ or 1 form $w=(2x^2y^2+6xy^3)dx + (8x^2y+x^2y^2)dy$.

I know that by definition, if there is an $f$ for which $df=w$, then for the first example, $f$ has to be 1 form, and for the second, $f$ has to be 0 form. But in practice, I am confused how to actually compute and get such $f$.

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Every exact form must be closed, so you can always check that first. If it turns out to be closed you try to find such an $f$ by integration. For example, for $w$ you need $f$ s.t. $\partial_x f= w_1$ and $\partial_y f=w_2$. So first integrate $w_1$ and adjust constant so that the second condition is also satisfied. – erlking Dec 14 '12 at 17:38

The second form is not closed, therefore not exact. The first one is exact. In fact if you take $$\eta=2yz dx+ xy dz$$ then $d\eta=w$. To get the 1-form $\eta $ you have to solve some differential equations. Let $\eta=f dx +g dy +h dz$. Then you need $$\frac{ \partial h }{\partial y }- \frac{ \partial g}{\partial z}=x$$ etc...

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