Compactly supported differential form

The exterior derivative maps $\Omega_c^k(M)$ to $\Omega_c^{k+1}(M)$ because $\operatorname{supp}(d\omega)\subseteq\operatorname{supp}\omega$. If $\eta$ is any smooth form, then $\eta\wedge\omega$ is compactly supported. Pullback along a smooth map need not preserve compact support, but pullback along a proper smooth map does.

On an oriented $n$-manifold, every $\omega\in\Omega_c^n(M)$ has a well-defined finite integral, even when $M$ is noncompact. The complex $(\Omega_c^\bullet(M),d)$ defines compactly supported de Rham cohomology. Compact support is also the hypothesis that removes contributions “at infinity” in the compact-support version of Stokes' theorem; see Bott and Tu, chapter on de Rham theory.

If $\varphi$ is a bump function and $\eta$ is any differential form, then $\varphi\eta$ is compactly supported. The standard volume form on $\mathbb R^n$ is not compactly supported. On the manifold $M=(0,1)$, the form $dx$ is not compactly supported: its support is all of $M$, which is not compact even though it is bounded as a subset of $\mathbb R$.

1. Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: de Rham theory, integration, and compact supports.
2. John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: differential forms, integration on manifolds, and de Rham cohomology.