Definition:Differential Form

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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 its natural structure as a $C^0$ manifold.

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A differential $k$-form is a continuous mapping:

$\omega : M \to \Lambda^kT^* M$

satisfying:

$\forall p \in M \map {\paren {\pi \circ \omega} } p = p$

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 mapping $\omega$ that assigns each point $p \in M$ an alternating $k$-form $\map \omega p$ on $T_p M$.

This article, or a section of it, needs explaining. In particular: What has this to do with differentials? In other words: can the language of this be tightened up? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Also see

• Results about differential forms can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential form
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential form

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