Definition:Differential Form

Definition
Let $M$ be an $n$-dimensional $C^1$ manifold.

Let $\ds \Lambda^k T^* M = \bigcup_{p \mathop \in M} \set p \times \map {\Lambda^k} {T_p^*M}$, 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 $\map {\paren {\pi \circ \omega} } p = p$ for all $p \in M$, where $\pi : \Lambda^k T^*M \to M$ is the projection onto the first argument, defined by $\map \pi {p, v} = p$.

In other words, a differential form is a continuous map $\omega$ that assigns each point $p \in M$ an alternating $k$-form $\map \omega p$ on $T_p M$.