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Formally, a degree $k$ differential form on a smooth manifold $M$ is an element of $\Omega^k(M) = \Gamma(M, \bigwedge^kT^*M)$ which is the vector space of smooth sections of the vector bundle $\pi: \bigwedge^kT^*M \to M$; a section is a map $\alpha : M \to \bigwedge^kT^*M$ such that $\pi\circ\alpha = \operatorname{id}_M$. In particular, if $\alpha \in \Omega^k(M)$, for each $x \in M$, $\alpha(x) \in \bigwedge^kT^*_xM$; that is, $\alpha(x)$ is an alternating map $(T^*_xM)^k \to \mathbb{R}$.