Editing Jet bundle (section)

==Vector fields==
A general [[vector field]] on the total space ''E'', coordinated by <math>(x, u) \mathrel\stackrel{\mathrm{def}}{=} \left(x^i, u^\alpha\right)\,</math>, is

:<math>V \mathrel\stackrel{\mathrm{def}}{=} \rho^i(x, u)\frac{\partial}{\partial x^i} + \phi^{\alpha}(x, u)\frac{\partial}{\partial u^\alpha}.\,</math>

A vector field is called '''horizontal''', meaning that all the vertical coefficients vanish, if <math>\phi^\alpha</math> = 0.

A vector field is called '''vertical''', meaning that all the horizontal coefficients vanish, if ''ρ<sup>i</sup>'' = 0.

For fixed ''(x, u)'', we identify

:<math>V_{(x, u)} \mathrel\stackrel{\mathrm{def}}{=} \rho^i(x, u) \frac{\partial}{\partial x^i} + \phi^{\alpha}(x, u) \frac{\partial}{\partial u^\alpha}\,</math>

having coordinates ''(x, u, ρ<sup>i</sup>, φ<sup>α</sup>)'', with an element in the fiber ''T<sub>xu</sub>E'' of ''TE'' over ''(x, u)'' in ''E'', called '''a [[tangent vector]] in ''TE'''''. A section

:<math>\begin{cases}
  \psi : E \to TE \\
    (x, u) \mapsto \psi(x, u) = V
\end{cases}</math>

is called '''a vector field on ''E''''' with

:<math>V = \rho^i(x, u) \frac{\partial}{\partial x^i} + \phi^\alpha(x, u) \frac{\partial}{\partial u^\alpha}</math>

and ψ in ''Γ(TE)''.

The jet bundle ''J<sup>r</sup>(π)'' is coordinated by <math>(x, u, w) \mathrel\stackrel{\mathrm{def}}{=} \left(x^i, u^\alpha, w_i^\alpha\right)\,</math>.  For fixed ''(x, u, w)'', identify

:<math>
  V_{(x, u, w)} \mathrel\stackrel{\mathrm{def}}{=}
    V^i(x, u, w) \frac{\partial}{\partial x^i} +
      V^\alpha(x, u, w) \frac{\partial}{\partial u^\alpha} +
      V^\alpha_i(x, u, w) \frac{\partial}{\partial w^\alpha_i} +
      V^\alpha_{i_1 i_2}(x, u, w) \frac{\partial}{\partial w^\alpha_{i_1 i_2}} + \cdots +
      V^\alpha_{i_1 \cdots i_r}(x, u, w) \frac{\partial}{\partial w^\alpha_{i_1 \cdots i_r}}
</math>

having coordinates

:<math>\left(x, u, w, v^\alpha_i, v^\alpha_{i_1 i_2}, \cdots, v^\alpha_{i_1 \cdots i_r}\right),</math>

with an element in the fiber <math>T_{xuw}(J^r\pi)</math> of ''TJ<sup>r</sup>(π)'' over ''(x, u, w)'' ∈ ''J<sup>r</sup>(π)'', called '''a tangent vector in ''TJ<sup>r</sup>(π)'''''. Here,

:<math>v^\alpha_i, v^\alpha_{i_1 i_2}, \ldots, v^\alpha_{i_1 \cdots i_r}</math>

are real-valued functions on ''J<sup>r</sup>(π)''. A section

:<math>\begin{cases}
  \Psi : J^r(\pi) \to  TJ^r(\pi) \\
        (x, u, w) \mapsto \Psi(u, w) = V
\end{cases}</math>

is '''a vector field on ''J<sup>r</sup>(π)''''', and we say <math>\Psi \in \Gamma(T\left(J^r\pi\right)).</math>