How can I tell if differential forms are exact or closed?
Let's take for example
$$i) \quad \omega = \frac12 dx -\frac1y dy -\frac2xdz$$
where $x,y,z \in \mathbb{R}^{+}$ and
$$ii) \quad \eta = x^2 dy \wedge dz +yx dz \wedge dx+z^3 dx \wedge dy $$
What definition do you use and what does it mean?
Any help very appreciated !
A differential form $p$-form $\omega$ is closed if $d\omega = 0$ and exact if there is a $(p-1)$-form $\eta$ such that $d\eta = \omega$. From these definitions it can be seen that every exact form is closed, but the converse is not generally true. Now using your definitions for $\omega$ and $\eta$: \begin{eqnarray} d\omega&=&\frac{1}{2}d(dx)+\frac{1}{y^2}dy\wedge dy+\frac{2}{x^2}dx\wedge dz=\frac{2}{x^2}dx\wedge dz, \\ d\eta &=& 2xdx\wedge dy \wedge dz +xdy\wedge dz\wedge dx+3z^2dz\wedge dx\wedge dy=(2x+ 3z^2)dx\wedge dy\wedge dz. \end{eqnarray} Then $\omega$ and $\eta$ are not closed.