# 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**

This article is complete as far as it goes, but it could do with expansion.In particular: There's plenty more in the above source in this entry which needs to be explored by someone who knows what they are doing hereYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.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 `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |