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
Famous quotes containing the word fields:
“Like a man traveling in foggy weather, those at some distance before him on the road he sees wrapped up in the fog, as well as those behind him, and also the people in the fields on each side, but near him all appears clear, though in truth he is as much in the fog as any of them.”
—Benjamin Franklin (17061790)