Math 6510 • Differentiable Manifolds

Let \(\omega\) be a compactly supported \(n\)-form on \(\mathbb{R}^n\), \(n\ge 1\). Show that the following statements
are equivalent:

\begin{enumerate}

\item

\[
\int_{\mathbb{R}^n}\omega=0.
\]



\item
There exists a compactly supported \((n-1)\)-form \(\eta\) on \(\mathbb{R}^n\) such that \(\omega=d\eta\).

You are allowed to use computations of de Rham cohomology of Euclidean spaces and
spheres but not of the compactly supported cohomology of \(\mathbb{R}^n\).

\end{enumerate}