Vector-valued differential form: Difference between revisions

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==Formal definition==

Let ''M'' be a [[smooth manifold]] and ''E'' → ''M'' be a smooth [[vector bundle]] over ''M''. We denote the space of [[section (fiber bundle)|smooth section]]s of a bundle ''E'' by Γ(''E''). An '''''E''-valued differential form''' of degree ''p'' is a smooth section of the [[tensor product]] bundle]] of ''E'' with Λ^''p''(''T''*''M''), the ''p''-th [[exterior power]] of the [[cotangent bundle]] of ''M''. The space of such forms is denoted by

:<math>\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M).</math>

Because Γ is a [[monoidal functor]],<ref name=gamma_monoidal>{{cite web|title=Global sections of a tensor product of vector bundles on a smooth manifold|url=http://math.stackexchange.com/questions/492166/global-sections-of-a-tensor-product-of-vector-bundles-on-a-smooth-manifold|website=math.stackexchange.com|accessdate=27 October 2014|ref=monoidal}}</ref> this can also be interpreted as