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 M is a smooth map
such that the image of x, denoted Vx, lies in TxM, the tangent space at x. In the language of fiber bundles, such a map is called a section. A vector field on M is therefore a section of the tangent bundle of M.
The set of all vector fields on M is denoted by Γ(TM). Vector fields can be added together pointwise
and multiplied by smooth functions on M
to get other vector fields. The set of all vector fields Γ(TM) then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted C∞(M).
A local vector field on M is a local section of the tangent bundle. That is, a local vector field is defined only on some open set U in M and assigns to each point of U a vector in the associated tangent space. The set of local vector fields on M forms a structure known as a sheaf of real vector spaces on M.
Read more about this topic: Tangent Bundle
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