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}$should give the ordinary integral and Stokes' theorem. Sorry if the answer is too obvious; I just don't have any of my textbooks available at the moment, and have never thought about Stokes' theorem for anything other than scalar-valued forms before. 2 Equipped$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}$should give the ordinary integral and Stokes' theorem. Sorry if the answer is too obvious; I just don't have any of my textbooks available at the moment, and have never thought about Stokes' theorem for anything other than scalar-valued forms before. 1 # Integration and Stokes' theorem for vector bundle-valued differential forms? Is there a version of Stokes' theorem for vector bundle-valued (or just vector-valued) differential forms? Concretely: Let$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.