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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2006 • Question 6

Question

Math 6510 • Differentiable Manifolds Winter 2006 • Question 6

Let \(\omega \) be a compactly supported smooth \(n\)-form on \(\mathbb {R}^n\). Prove that the following two statements are equivalent.

  1. There exists a compactly supported smooth \((n-1)\)-form \(\eta \) such that \(\omega =d\eta \).
  2. \(\displaystyle \int _{\mathbb {R}^n}\omega =0\).

Note: You are allowed to use the fact that \(H^n(S^n)=\mathbb {R}\), but you are not allowed to quote the fact that \(H_c^n(\mathbb {R}^n)=\mathbb {R}\).

Raw LaTeX

Let \(\omega\) be a compactly supported smooth \(n\)-form on \(\mathbb{R}^n\). Prove
that the following two statements are equivalent.
\begin{enumerate}
\item There exists a compactly supported smooth \((n-1)\)-form \(\eta\) such that
\(\omega=d\eta\).
\item \(\displaystyle\int_{\mathbb{R}^n}\omega=0\).
\end{enumerate}
Note: You are allowed to use the fact that \(H^n(S^n)=\mathbb{R}\), but you are not
allowed to quote the fact that \(H_c^n(\mathbb{R}^n)=\mathbb{R}\).

Topic Tags

Cohomology and Cup Products Compactly Supported Forms and Integration Differential Forms and Exterior Calculus de Rham Cohomology