Editing Connection (vector bundle) (section)

==Formal definition==
Let <math>E\to M</math> be a smooth real [[vector bundle]] over a [[smooth manifold]] <math>M</math>. Denote the space of smooth [[Section (fiber bundle)|sections]] of <math>E\to M</math> by <math>\Gamma(E)</math>. A '''covariant derivative''' on <math>E\to M</math> is either of the following equivalent structures:
# an <math>\mathbb{R}</math>-[[linear map]] <math>\nabla : \Gamma(E) \to \Gamma(T^*M\otimes E)</math> such that the [[product rule]] <math display="block">\nabla(fs) = df\otimes s + f\nabla s</math> holds for all [[smooth functions]] <math>f</math> on <math>M</math> and all smooth sections <math>s</math> of <math>E.</math>
# an assignment, to any smooth section {{mvar|s}} and every <math>x\in M</math>, of a <math>\mathbb{R}</math>-linear map <math>(\nabla s)_x:T_xM\to E_x</math> which depends smoothly on {{mvar|x}} and such that <math display="block">\nabla(a_1s_1+a_2s_2)=a_1\nabla s_1+a_2\nabla s_2</math> for any two smooth sections <math>s_1,s_2</math> and any real numbers <math>a_1,a_2,</math> and such that for every smooth function <math>f</math>, <math>\nabla(fs)</math> is related to <math>\nabla s</math> by <math display="block">\big(\nabla(fs)\big)_x(v)=df(v)s(x)+f(x)(\nabla s)_x(v)</math> for any <math>x\in M</math> and <math>v\in T_xM.</math>
Beyond using the canonical identification between the vector space <math>T_x^\ast M\otimes E_x</math> and the vector space of linear maps <math>T_xM\to E_x,</math> these two definitions are identical and differ only in the language used.

It is typical to denote <math>(\nabla s)_x(v)</math> by <math>\nabla_vs,</math> with <math>x</math> being implicit in <math>v.</math> With this notation, the product rule in the second version of the definition given above is written
:<math>\nabla_v(fs)=df(v)s+f\nabla_vs.</math>

''Remark.'' In the case of a complex vector bundle, the above definition is still meaningful, but is usually taken to be modified by changing "real" and "<math>\mathbb{R}</math>" everywhere they appear to "complex" and "<math>\mathbb{C}.</math>" This places extra restrictions, as not every real-linear map between complex vector spaces is complex-linear. There is some ambiguity in this distinction, as a complex vector bundle can also be regarded as a real vector bundle.