Vector field on a manifold A section of the tangent bundle . The set of differentiable vector fields forms a module over the ring of differentiable functions on .

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Example 1.

For a chart of the manifold one defines the -th basic vector field according to the formula where is the -th basic tangent vector to at the point . Any vector field can be uniquely represented in the form where are the components of in . Since a vector field can be regarded as a derivation of the ring (see example 2), the set of vector fields forms a Lie algebra with respect to the commutation operation (the Lie bracket).

Example 2.

For the chart and , the function is defined by the formula  where is the partial derivative with respect to . Note that ; is called the derivative of in the direction .

Example 3.

For the chart and , the commutator (Lie bracket) of the vector fields is defined by the formula  It satisfies the relations  in particular, Each vector field induces a local flow on — a family of diffeomorphisms of a neighbourhood , such that for and is the integral curve of the vector field through , i.e. where is the tangent vector to at . Conversely, a vector field is associated with a local flow , which is a variation of the mapping ; here Each vector field defines a Lie derivation of a tensor field of type with values in a vector space (infinitesimal transformation of ), corresponding to the local flow ; its special cases include the action of the vector field on , and the Lie bracket A vector field without singularities generates an integrable one-dimensional differential system as well as a Pfaffian system associated with it on .

A generalization of the concept of a vector field on a manifold is that of a vector field along a mapping , which is a section of the bundle induced by , as well as a tensor field of type , which is a section of the bundle associated with with the aid of the functor .