Editing Tangent bundle (section)

==Vector fields==
A smooth assignment of a tangent vector to each point of a manifold is called a '''[[vector field]]'''. Specifically, a vector field on a manifold <math> M </math> is a [[smooth map]]
:<math>V\colon M \to TM</math>

such that <math>V(x) = (x,V_x)</math> with <math>V_x\in T_xM</math> for every <math>x\in M</math>. In the language of fiber bundles, such a map is called a ''[[section (fiber bundle)|section]]''. A vector field on <math>M</math> is therefore a section of the tangent bundle of <math>M</math>.

The set of all vector fields on <math>M</math> is denoted by <math>\Gamma(TM)</math>. Vector fields can be added together pointwise

:<math>(V+W)_x = V_x + W_x</math>
and multiplied by smooth functions on ''M''

:<math>(fV)_x = f(x)V_x</math>

to get other vector fields. The set of all vector fields <math>\Gamma(TM)</math> then takes on the structure of a [[module (mathematics)|module]] over the [[associative algebra|commutative algebra]] of smooth functions on ''M'', denoted <math>C^{\infty}(M)</math>.

A local vector field on <math>M</math> is a ''local section'' of the tangent bundle. That is, a local vector field is defined only on some open set <math>U\subset M</math> and assigns to each point of <math>U</math> a vector in the associated tangent space. The set of local vector fields on <math>M</math> forms a structure known as a [[sheaf (mathematics)|sheaf]] of real vector spaces on <math>M</math>.

The above construction applies equally well to the cotangent bundle – the differential 1-forms on <math>M</math> are precisely the sections of the cotangent bundle <math>\omega \in \Gamma(T^*M)</math>, <math>\omega: M \to T^*M</math> that associate to each point <math>x \in M</math> a 1-covector <math>\omega_x \in T^*_xM</math>, which map tangent vectors to real numbers:  <math>\omega_x : T_xM \to \R</math>. Equivalently, a differential 1-form <math>\omega \in \Gamma(T^*M)</math> maps a smooth vector field <math>X \in \Gamma(TM)</math> to a smooth function <math>\omega(X) \in C^{\infty}(M)</math>.