# Definition:Differential Form

Jump to navigation
Jump to search

This article needs to be tidied.Please fix formatting and $\LaTeX$ errors and inconsistencies. It may also need to be brought up to our standard house style.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 `{{Tidy}}` from the code. |

## 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.

This article, or a section of it, needs explaining.In particular: What is $\map {\Lambda^k} {T_p^*M}$?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. |

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 $T_p M$.

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. |

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.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 `{{MissingLinks}}` from the code. |