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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