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A $V$-valued differential $n$-form $\omega$ on a manifold $M$ is a section of the bundle $\Lambda^n (T^*M) \otimes V$. (That is, the restriction $\omega_p$ to any tangent space $T_p M$ for $p \in M$ is a completely antisymmetric map $\omega_p : T_p M \times T_p M \times \cdots \times T_p M \to V$.) $V$ is a vector space here.

One can define a flat covariant derivative $\mathrm{d}\colon \Lambda^n (T^*M) \to \Lambda^{n+1} (T^*M)$ which is just the exterior derivative. It fulfills Stokes' theorem.

Assume now an algebra structure on $V$, and a representation $\rho$ of $V$ on a vector space $W$. For a chosen $V$-valued differential 1-form $\omega$, there is also a covariant derivative (like a principal connection) that acts on all $W$-valued differential forms $\phi$ by the formula $\mathrm{d}_\omega \phi := \mathrm{d} \phi + \omega \wedge_\rho \phi$. The product $\wedge_\rho$ is the composition of $\wedge$, which multiplies a $V$-valued $n$-form and a $W$-valued $m$-form to a $V \otimes W$-valued $(n+m)$-form, and $\rho$.

Is there a generalisation for Stokes' theorem for $\mathrm{d}_\omega$? Maybe something like $\int_M \mathrm{d}_\omega \phi = \int_{\partial M} \phi$ up to terms proportional to the curvature of $\mathrm{d}_\omega$?

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    I would like to tag this question with "principal-connection" and "stokes-theorem", which are new tags. If there is someone with high reputation who finds this appropriate, could he/she tag my question accordingly?2011-07-05
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    I think these two tags are far too specific to be of real use.2011-07-05
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    There are $\sim 10^1$ questions about Stokes theorem, I think that justifies a tag.2011-07-05
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    As I said, I think it's too specific and (differential-forms) is a good enough match in my opinion, so I'm not going to create this tag. But if someone else wants to, feel free to do so.2011-07-05
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    I agree with Theo. If I see a (stokes-theorem) tag I would just merge it into (differential-forms) or (differential-geometry). Ditto (principal-connection). Questions about "Stokes Theorem" or "Principal Connections" would almost certainly have those words in the question tag anyway, so a tag is somewhat redundant for searching in those cases.2011-07-05
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    Very interesting question, BTW. The usual generalization of Stokes' theorem to the case of a principal connection is in the context of non-Abelian gauge theories. The generalization, however, is to generalize the relation between holonomy and curvature. (For example, [this](http://arxiv.org/abs/math-ph/0012035).) If you don't get an answer here after a few days, I encourage to ask on MathOverflow.2011-07-05
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    The paper you pointed at is quite interesting, it shows the theorem for $\phi = A$ in two dimensions.2011-07-05
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    There was a similar question on MO earlier, with a complete answer http://mathoverflow.net/questions/35334/integration-and-stokes-theorem-for-vector-bundle-valued-differential-forms2011-10-24
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    @Yuri: Thank you very much, that was insightful.2011-10-26
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    @Jesse: I don't understand your question. Read the first paragraph of my question, there you have the definition.2011-10-26
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    @Turion: Nevermind my comment (now deleted). Even I'm not sure what I meant.2011-10-27
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    The question could need some clarification. What are $V$ and $W$? Vector spaces, vector bundles, or something else? And $\omega:TM\rightarrow V$ would be a $V$ valued 1-form, while $\Lambda^n(T_pM,V)$ is an $n$-form. Is $n$ the dimension of $M$ or any number?2017-08-11
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    @EinarRødland, I've reworked the question, I hope it's clearer now.2017-08-11

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