show/hide this revision's text 3 I meant of course connection on $E$, not on $M$.

Is there a version of Stokes' theorem for vector bundle-valued (or just vector-valued) differential forms?

Concretely: Let $E \rightarrow M$ be a smooth vector bundle over an $n$-manifold $M$ equipped with a connection. First of all, is there an $E$-valued integration defined on the space $\Omega^{n-1}(M,E)$ of smooth sections of $E\otimes \Lambda^{n-1}T^\ast M$ mimicking what you have for $\mathbb{R}$-valued forms? If so, does

$\int_{\partial M} \omega = \int_M d\omega$

hold (or even make sense) for $\omega\in\Omega^{n-1}(M,E)$ if $d$ is the exterior derivative coming from our connection on $M$?E$?

In this setting, letting $E$ be the trivial bundle $M\times \mathbb{R}$ should give the ordinary integral and Stokes' theorem.

Sorry if the answer is too obvious; I just don't have any of my textbooks available at the moment, and have never thought about Stokes' theorem for anything other than scalar-valued forms before.
show/hide this revision's text 2 Equipped $M$ with a connection. Thanks to skupers' comment.

Is there a version of Stokes' theorem for vector bundle-valued (or just vector-valued) differential forms?

Concretely: Let $E \rightarrow M$ be a smooth vector bundle over an $n$-manifold $M$. M$ equipped with a connection. First of all, is there an $E$-valued integration defined on the space $\Omega^{n-1}(M,E)$ of smooth sections of $E\otimes \Lambda^{n-1}T^\ast M$ mimicking what you have for $\mathbb{R}$-valued forms? If so, does

$\int_{\partial M} \omega = \int_M d\omega$

hold (or even make sense) for $\omega\in\Omega^{n-1}(M,E)$?\omega\in\Omega^{n-1}(M,E)$ if $d$ is the exterior derivative coming from our connection on $M$?

In this setting, letting $E$ be the trivial bundle $M\times \mathbb{R}$ should give the ordinary integral and Stokes' theorem.

Sorry if the answer is too obvious; I just don't have any of my textbooks available at the moment, and have never thought about Stokes' theorem for anything other than scalar-valued forms before.
show/hide this revision's text 1

Integration and Stokes' theorem for vector bundle-valued differential forms?

Is there a version of Stokes' theorem for vector bundle-valued (or just vector-valued) differential forms?

Concretely: Let $E \rightarrow M$ be a smooth vector bundle over an $n$-manifold $M$. First of all, is there an $E$-valued integration defined on the space $\Omega^{n-1}(M,E)$ of smooth sections of $E\otimes \Lambda^{n-1}T^\ast M$ mimicking what you have for $\mathbb{R}$-valued forms? If so, does

$\int_{\partial M} \omega = \int_M d\omega$

hold (or even make sense) for $\omega\in\Omega^{n-1}(M,E)$?

In this setting, letting $E$ be the trivial bundle $M\times \mathbb{R}$ should give the ordinary integral and Stokes' theorem.

Sorry if the answer is too obvious; I just don't have any of my textbooks available at the moment, and have never thought about Stokes' theorem for anything other than scalar-valued forms before.