Vector Fields
A general vector field on the total space E, coordinated by, is
A vector field is called horizontal, meaning all the vertical coefficients vanish, if φα = 0.
A vector field is called vertical, meaning all the horizontal coefficients vanish, if ρi = 0.
For fixed (x, u), we identify
having coordinates (x, u, ρi, φα), with an element in the fiber TxuE of TE over (x,u) in E, called a tangent vector in TE. A section
is called a vector field on E' with
and ψ in Γ(TE).
The jet bundle Jr(π) is coordinated by . For fixed (x,u,w), identify
having coordinates, with an element in the fiber of TJr(π) over (x, u, w) ∈ Jr(π), called a tangent vector in TJr(π). Here,
are real-valued functions on Jr(π). A section
is a vector field on Jr(π), and we say .
Read more about this topic: Jet 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)