#### Problem 6

Is $\alpha $ $\left( 3a\,db\wedge dc\right)$ closed? Exact? What about $\beta $ $\left( \sin \theta \,d\theta \wedge d\phi \right)$ and $\gamma $ $\left( -\sin \theta \cos ^{2}\theta d\theta +\sin \theta d\phi \right)$?

#### Solution

For the closedness, we should check whether $d(.)=0$;

(1)Thus, only $\beta$ is found to be closed.

For the exactness, we should check if $\exists \eta$ $k-1$ form such that $(.)=d\eta$ where $(.)$ is a $k-form$;

If $\alpha$ is exact, then a candidate $1-form$, $\eta$ should have either $db$ or $dc$ in it. Let $\eta =f(a)db$, then

(4)Let $\eta =f(a,c)db$, then

(5)Similar argument is valid for $\eta =f(.)dc$. Therefore, $\alpha$ is not exact.

For $\beta$, let $\eta =-\cos \theta d\phi$, then

(6)Hence, $\beta$ is exact.

For $\gamma$, $\eta =g(\theta ,\phi )$, so

(7)If $\gamma$ is exact, then;

(8)But, from the second equation, the function $g(\theta ,\phi )$ should be in the form of $g(\theta ,\phi )=(\phi + C) \sin \theta$ where $C$ is a constant.

If we plug that in to the first partial differential equation and solve the left hand side;

(10)If $(\phi + C) \cos \theta =-\sin \theta \cos ^{2}\theta \Rightarrow \phi =-\sin\theta \cos \theta - C$ which results in $1$ dimensional manifold with coordinate $\theta$.

Therefore, this is a contradiction to the actual dimension of the $2-$torus and concludes that $\gamma$ is not exact.

Scott agrees that only $\beta$ is closed and only $\beta$ is exact. Note, however, that a form can't be exact if

it's not closed, so more work was done above than was absolutely necessary.