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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“Genius is the naturalist or geographer of the supersensible regions, and draws their map; and, by acquainting us with new fields of activity, cools our affection for the old. These are at once accepted as the reality, of which the world we have conversed with is the show.”
—Ralph Waldo Emerson (1803–1882)