---- - [ ] right now this note is also collecting general vector-valued tensor fields > [!definition] Definition. ([[differential form with values in a vector bundle]]) > Let $E \xrightarrow{\pi}B$ be a (smooth) [[vector bundle]] over (smooth) $n$-[[smooth manifold|manifold]] $B$ with typical fiber $V$. Let $\Gamma(E)$ denote its space of [[vector bundle|smooth]] [[vector bundle|sections]]. > > A **differential form with values in $E$**, or **$E$-valued differential form**, of degree $r$ is (smooth) [[section]] of the [[tensor product of vector bundles|tensor product bundle]] $\Lambda^{r}T^{*}B \otimes E$.[^1] The space of such forms is denoted $\Omega_{B}^{r}(E):=\Gamma(\Lambda^{r}T^{*}B \otimes E)$. > > The name stems from the fact that, in a [[vector bundle|local trivialization]] $U$, a section $s \in \Omega_{B}^{r}(E)$ corresponds precisely to an assignment of an [[alternating multilinear map]] $s_{u} \in A(\mathbb{R}^{n},\dots, \mathbb{R}^{n};V)$ to each $u \in U$. In other words, $s$ is an assignment of a 'vector of differential $r$-forms' to each $u \in U$.[^2] The 'usual' [[differential form|differential forms]] arise in the trivial case where $V=\mathbb{R}$ and $U=B$ (so $E=B \times \mathbb{R}$ is the trivial line bundle). > > An important special case is $r=1$. The typical fiber of $T^{*}B \otimes E$ is $(\mathbb{R}^{n})^{*} \otimes V \cong \text{Hom}(\mathbb{R}^{n}, V)$, so a [[section]] $s \in \Omega_{B}^{1}(E)$ is an assignment of a 'vector of 1-forms' to each $u \in U$. > > We can also consider $\text{End }E$ instead of $E$. Differential $r$-forms with values in this bundle are (locally) most concretely viewed as *matrices* of differential $r$-forms (viewing $\text{End }V \cong \mathbb{R}^{m \times m}$). Note that we can 'do matrix algebra' with matrices and vectors of differential forms by replacing what is usually scalar multiplication with the [[algebra of alternating multilinear forms|wedge product]]. > [!definition] Definition. ([[differential form with values in a vector bundle]]) Let $E \xrightarrow{\pi}B$ be a (smooth) [[vector bundle]] over (smooth) $n$-[[smooth manifold|manifold]] $B$ with typical fiber $V$. Let $\Gamma(E)$ denote its space of [[vector bundle|smooth]] [[vector bundle|sections]]. A **(differential) $r$-form with values in $E$**, or **$E$-valued (differential) $r$-form**, is a (smooth) [[section]] of the [[tensor product of vector bundles|tensor product bundle]] $\Lambda^{r}T^{*}B \otimes E$.[^1] The space of such forms is denoted $\Omega_{B}^{r}(E):=\Gamma(\Lambda^{r}T^{*}B \otimes E)$. An $r$-form $s \in \Omega^{r}_{B}(E)$ corresponds to a (smoothly varying) assignment of an [[alternating multilinear map]] $s_{b} \in A(\overbrace{ T^{}_{b}B,\dots,T^{}_{b}B }^{ r \text{ times} }; E_{b})$ to each $b \in B$. In other words, $s$ is an assignment of a 'vector of differential $r$-forms' to each $b \in B$.[^2] In a simultaneous [[vector bundle|local trivialization]] + [[coordinate chart|coordinate neighborhood]] $U$, $s \in \Omega^{r}_{B}(E)$ looks like a [[linear combination]] $s |_{U}=\sum_{i_{1} < \dots < i_{r}, j} s^{j}_{i_{1} \cdots i_{r}}(dx^{i_{1}} \wedge \dots \wedge dx^{i_{r}} \otimes e_{j})$ for $s_{i_{1} \cdots i_{r}}^{j} \in C^{\infty}(U)$, where $\{ e_{j}(p) \}_{j=1}^{\text{rank }E}$, $e_{j} \in \Gamma(E, U)$, form a smoothly varying basis of the fibers $E_{p}$. An important special case is $r=1$. The typical fiber of $T^{*}B \otimes E$ is $(\mathbb{R}^{n})^{*} \otimes V \cong \text{Hom}(\mathbb{R}^{n}, V)$, so a [[section]] $s \in \Omega_{B}^{1}(E)$ is an assignment of a 'vector of 1-forms' to each $u \in U$. We can also consider $\text{End }E$ instead of $E$. Differential $r$-forms with values in this bundle are (locally) most concretely viewed as *matrices* of differential $r$-forms (viewing $\text{End }V \cong \mathbb{R}^{m \times m}$). Note that we can 'do matrix algebra' with matrices and vectors of differential forms by replacing what is usually scalar multiplication with the [[algebra of alternating multilinear forms|wedge product]]. [^1]: Here $T^{*}B$ is the [[cotangent bundle]] of $B$ and $\Lambda^{r}T^{*}B$ is its $r$th [[exterior power]]. The [[vector bundle]] $\Lambda^{r}T^{*}B \otimes E$ has (by construction) typical fiber $\Lambda^{r}\big( (\mathbb{R}^{n})^{*} \big) \otimes V \cong A( \overbrace{\mathbb{R}^{n}, \dots, \mathbb{R}^{n}}^{r \text{ times}}; V )$. [^2]: If it helps, imagine $E_{b} \cong V \cong \mathbb{R}^{k}$, $T_{p}B \cong \mathbb{R}^{n}$ so that $s_{u}:(\mathbb{R}^{n})^{r} \to E_{b}$ has components $s_{u}=(s_{u_{1}},\dots,s_{u_{k}})$. Each component is an [[alternating multilinear map]] $(\mathbb{R}^{n})^{r} \to \mathbb{R}$. ---- #### ![[vector bundle data table]] See also: https://en.wikipedia.org/wiki/Vector-valued_differential_form#:~:text=In%20mathematics%2C%20a%20vector%2Dvalued,as%20R%2Dvalued%20differential%20forms. ---- #### 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 > ```