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Einstein convention in effect.
> [!definition]
> If $\omega \in \Omega^{\ell}(B)$ is an ordinary $\ell$-form and $\boldsymbol \sigma \in \Omega_{B}^{q}(E)$ is an $E$-valued $q$-form, we define their **wedge product** $\boldsymbol \sigma \wedge \omega$ to be the $E$-valued $(q+\ell)$-form defined at $p \in B$ by
>
> $(\boldsymbol \sigma \wedge \omega)_{p}=\big(\eta_{p} \wedge \omega_{p} \big) \otimes e_{p} \in \Lambda^{q+\ell}_{}T^{*}_{p}B \otimes E_{p}$
>
> for pure tensors $\boldsymbol \sigma_{p}=\eta_{p} \otimes e_{p} \in (\Lambda^{q} T^{*}B \otimes E)_{p}$ and extending linearly.
> [!definition] Definition. ([[higher covariant derivative]])
> Let $E \xrightarrow{\pi}B$ be a [[vector bundle]]. The definition in [[covariant derivative on a vector bundle|covariant derivative]] applies to [[vector bundle|sections]], i.e., [[differential form with values in a vector bundle|vector-valued]] $0$-forms. It may be extended, *[[three views on connections|with the help of a]] [[connection on a vector bundle|connection]]* $A$, to work with [[differential form with values in a vector bundle|vector-valued]] forms of any degree. The object we require is a graded $\mathbb{R}$-linear operator $d_{A}:\Omega_{B}^{r}(E) \to \Omega_{B}^{r+1}(E)$ satisfying a [[derivation|Leibniz rule]] analogous to that for ordinary [[differential form|differential forms]]: $d_{A}(\sigma \wedge \omega) = (d_{A}\sigma) \wedge \omega + (-1)^{q}\sigma \wedge d \omega,$
where ($r=q+\ell$)
>- $\sigma \in \Omega_{B}^{q}(E)$ is a [[differential form with values in a vector bundle|differential]] $q$-[[differential form with values in a vector bundle|form]] valued in $E$, and is analogous to $s \in \Omega^{0}_{B}(E)$ in the [[covariant derivative on a vector bundle|original definition]];
>- $\omega \in \Omega^{\ell}(B)$ is an ordinary [[differential form]] on $B$, analogous to $f \in C^{\infty}(B)=\Omega^{0}(B)$ in the [[covariant derivative on a vector bundle|original definition]].
>
Analogous to scalar-vector multiplication, the notation $\sigma \wedge \omega$ is understood to happen elementwise (viewing $\sigma$ as a [[vector-valued|vector of]] ordinary [[differential form|differential forms]]).
>
More explicitly, in a [[vector bundle|local trivialization]] (and also [[coordinate chart|coordinate neighborhood]]) of $E$; then one can compute $d_{A}(\sigma)=d\sigma + A \wedge \sigma$ for $s_{I} \in \Gamma(E |_{U})=\Omega^{0}_{U}(E |_{U})$ and $\sigma \in \Omega_{U}^{q}(E |_{U})$.
^definition
> [!basicexample]
> Suppose $r=1$. Let $f \in C^{\infty}(B)=\Omega^{0}(B)$, $\omega \in \Omega^{1}(B)$, $\boldsymbol \sigma \in \Omega^{1}_{B}(E)$. The Leibniz rule in this case consists of two conditions $(1+1=2, 2+0=2)$, namely:
> 1. $d_{A}(f\boldsymbol \sigma )=f d_{A} \boldsymbol \sigma +\boldsymbol \sigma \wedge df$
> 2. $d_{A}(\boldsymbol \sigma \wedge \omega)=(d_{A}\boldsymbol \sigma) \wedge \omega + \boldsymbol \sigma \wedge d\omega$.
>
>
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Application to the [[endomorphism bundle]] $\text{End }E$ constructs an operator $d_{A}:\Omega_{B}^{r}(\text{End }E)\to \Omega_{B}^{r+1}(\text{End }E)$that provides a notion of covariantly differentiating general "matrix-valued differential forms".
- [ ] actually, there is more going on here
> [!justification] That computation.
>
in coordinates one has by Leibniz rule:[^1] assuming $\sigma=s_{I}dx^{I} \in \Omega^{q}_{B}(E)$,
$d_{A}(\sigma)=d_{A}(s_{I} d x^{I})=(d_{A}s_{I}) \wedge dx^{I}.$
$d_{A}s_{I}$ is just the original [[covariant derivative on a vector bundle|covariant derivative]] of a section $s_{I} \in \Omega^{0}_{B}(E)=\Gamma(E)$; so we can write this as $d_{A}(\sigma)=(ds_{I}+As_{I}) \wedge dx^{I}=(ds_{I}) \wedge dx^{I} + (As_{I}) \wedge dx^{I}.$
Recognizing that $(ds_{I}) \wedge dx^{I}$ is precisely the definition of the [[exterior derivative]] $d(s_{I} dx^{I})=d(\sigma)$ (where everything is understood to 'happen elementwise' as relevant), and that $s_{I}$ and '$\wedge