Editing Connection form (section)

===Exterior connections===
{{main|Exterior covariant derivative}}
A [[connection (vector bundle)|connection]] in ''E'' is a type of [[differential operator]]
:<math>D : \Gamma(E) \rightarrow \Gamma(E\otimes T^*M) = \Gamma(E)\otimes\Omega^1M</math>
where Γ denotes the [[sheaf (mathematics)|sheaf]] of local [[section (fibre bundle)|sections]] of a vector bundle, and Ω<sup>1</sup>''M'' is the bundle of differential 1-forms on ''M''.  For ''D'' to be a connection, it must be correctly coupled to the [[exterior derivative]].  Specifically, if ''v'' is a local section of ''E'', and ''f'' is a smooth function, then
:<math>D(fv) = v\otimes (df) + fDv</math>
where ''df'' is the exterior derivative of ''f''.

Sometimes it is convenient to extend the definition of ''D'' to arbitrary [[vector-valued differential form|''E''-valued forms]], thus regarding it as a differential operator on the tensor product of ''E'' with the full [[exterior algebra]] of differential forms.  Given an exterior connection ''D'' satisfying this compatibility property, there exists a unique extension of ''D'':
:<math>D : \Gamma(E\otimes\Lambda^*T^*M) \rightarrow \Gamma(E\otimes\Lambda^*T^*M)</math>
such that
:<math> D(v\wedge\alpha) = (Dv)\wedge\alpha + (-1)^{\text{deg}\, v}v\wedge d\alpha</math>
where ''v'' is homogeneous of degree deg ''v''.  In other words, ''D'' is a [[derivation (abstract algebra)|derivation]] on the sheaf of graded modules Γ(''E'' ⊗ Ω<sup>*</sup>''M'').