Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 4

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 4

Let \(\omega \) be a compactly supported smooth \(n\)-form on \(\mathbb {R}^n\). Show that there exists a compactly supported smooth \((n-1)\)-form \(\eta \) with \(\omega =d\eta \) if and only if \[ \int _{\mathbb {R}^n}\omega =0. \] You are allowed to quote facts about \(H^k(X)\) but not about \(H_c^k(X)\).

Raw LaTeX

Let \(\omega\) be a compactly supported smooth \(n\)-form on \(\mathbb{R}^n\). Show
that there exists a compactly supported smooth \((n-1)\)-form \(\eta\) with \(\omega=d\eta\)
if and only if
\[
  \int_{\mathbb{R}^n}\omega=0.
\]
You are allowed to quote facts about \(H^k(X)\) but not about \(H_c^k(X)\).

Topic Tags

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