---- > [!definition] Definition. ([[differential form]]) > Let $M$ be a [[smooth manifold]] of dimension $\textcolor{Skyblue}{n}$. A (smooth) **(exterior) (differential) $r$-form** $\omega$, or just an **$r$-form on $M$**, is a (smooth) [[section]] of the $r$th [[exterior power]] $\Lambda^{r} T^{*}M$ of the [[cotangent bundle]] of $M$, that is, a smooth map $\alpha: M \to \Lambda^{r}T^{*}M$ such that $\alpha(p) \in \Lambda^{r}T_{p}^{*}M$. It can also be called an **alternating tensor field**. > The [[vector space]] of (smooth) differential $r$-forms on $M$ is denoted $\Omega^{r}(M)$. Elements of $\Omega^{\textcolor{Skyblue}{n}}(M)$ are called **top-degree forms**, or **volume forms**. By convention, $\Omega^{0}(M):=C^{\infty}(M)$. > [[coordinate chart|Local coordinates]] $\big(U, (x^{1},\dots,x^{n})\big)$ give rise to [[tangent space at a point of a smooth manifold|coordinate bases]] $\left( \frac{ \partial }{ \partial x^{i} } |_{p} \right)$ which [[dual basis|dualize]] into [[cotangent space|cotangent bases]] $(dx^{i} |_{p})$ for each $p \in U$. By considering the $1$-forms $p \xmapsto{dx^{i}} dx^{i} |_{p}$, it is evident that $\Omega^{r}(U)$ has a [[basis]] $(dx^{I})_{I \in \text{ASC}_{k,n}}$, so that [^1] $\omega |_{U}= \sum_{I \in \text{ASC}_{k,n}} \omega_{I} \ dx^{I}=\sum_{1 \leq i_{1} < \dots < i_{r} \leq n} \omega_{i_{1}\dots i_{r}} \ dx^{i_{1} } \wedge \dots \wedge dx^{i_{r}}$where the coefficients $\omega_{I}=\omega_{I}(p)$ are continuous scalar functions $U \to \mathbb{R}$, with $\omega_{I}=\omega\left( \frac{ \partial }{ \partial x^{i_{1}} } , \dots, \frac{ \partial }{ \partial x^{i_{r}} } \right)$ ^definition > [!note] Note. > TT has a nice exposition: https://www.math.ucla.edu/~tao/preprints/forms.pdf ^note [^1]: Where we recall that the $\wedge$ seen here comes from [[de Rham algebra of differential forms on a smooth manifold]], which defines a **wedge product of differential forms** $(\omega \wedge \eta)_{p}:=\omega_{p} \wedge \eta_{p}$ pointwise (the latter entity is the [[algebra of alternating multilinear forms|wedge product]] in the [[space and algebra of alternating tensors at a point on a manifold|algebra of alternating tensors]] at $p$) ---- #### - [ ] differential graded algebra $\Omega^{*}(M):=\bigoplus_{r=1?}^{\text{dim } M}\Omega^{r}(M)$ ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```