Definition:Differential Form

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Let $M$ be an $n$-dimensional $C^1$ manifold. Let $\Lambda^kT^*M = \bigcup_{p \in M}\{p\} \times \Lambda^k(T_p^*M)$, endowed with it's natural structure as a $C^0$ manifold. A differential $k$-form is a continuous map $\omega : M \to \Lambda^kT^*M$ satisfying $(\pi \circ \omega)(p) = p$ for all $p \in M$, where $\pi : \Lambda^kT^*M \to M$ is the projection onto the first argument, defined by $\pi(p, v) = p$. In other words, a differential form is a continuous map $\omega$ that assigns each point $p \in M$ an alternating $k$-form $\omega(p)$ on $T_pM$.