Editing Jet bundle (section)

==Jet manifolds==
The '''''r''-th jet manifold of π''' is the set

:<math>J^r (\pi) = \left \{j^r_p\sigma:p \in M, \sigma \in \Gamma(p) \right \}.</math>

We may define projections ''π<sub>r</sub>'' and ''π''<sub>''r'',0</sub> called the '''source and target projections''' respectively, by

:<math>\begin{cases}
  \pi_r: J^r(\pi) \to M \\
  j^r_p\sigma \mapsto p
\end{cases}, \qquad \begin{cases}
  \pi_{r, 0}: J^r(\pi) \to E \\
  j^r_p\sigma \mapsto \sigma(p)
\end{cases}</math>

If 1 ≤ ''k'' ≤ ''r'', then the '''''k''-jet projection''' is the function ''π<sub>r,k</sub>'' defined by

<math display=block>\begin{cases}
  \pi_{r, k}: J^r(\pi) \to J^{k}(\pi) \\
           j^r_p\sigma \mapsto j^{k}_p\sigma
\end{cases}</math>

From this definition, it is clear that ''π<sub>r</sub>'' = ''π'' <small> o </small> ''π''<sub>''r'',0</sub> and that if 0 ≤ ''m'' ≤ ''k'', then ''π<sub>r,m</sub>'' = ''π<sub>k,m</sub>'' <small> o </small> ''π<sub>r,k</sub>''. It is conventional to regard ''π<sub>r,r</sub>'' as the [[identity function|identity map]] on ''J&thinsp;<sup>r</sup>''(''π'') and to identify ''J''&thinsp;<sup>0</sup>(''π'') with ''E''.

The functions ''π<sub>r,k</sub>'', ''π''<sub>''r'',0</sub> and ''π<sub>r</sub>'' are [[Smooth function|smooth]] [[surjective]] [[submersion (mathematics)|submersion]]s.

[[File:Jet Bundle Image FbN.png|500px|center]]

A [[coordinate system]] on ''E'' will generate a coordinate system on ''J&thinsp;<sup>r</sup>''(''π''). Let (''U'', ''u'') be an adapted [[coordinate chart]] on ''E'', where ''u'' = (''x<sup>i</sup>'', ''u<sup>α</sup>''). The '''induced coordinate chart (''U<sup>r</sup>'', ''u<sup>r</sup>'')''' on ''J&thinsp;<sup>r</sup>''(''π'') is defined by

<math display=block>\begin{align}
  U^r &= \left\{j^r_p \sigma: p \in M, \sigma(p) \in U\right\} \\
  u^r &= \left(x^i, u^\alpha, u^\alpha_I\right)
\end{align}</math>

where

<math display=block>\begin{align}
       x^i\left(j^r_p\sigma\right) &= x^i(p) \\
  u^\alpha\left(j^r_p\sigma\right) &= u^\alpha(\sigma(p))
\end{align}</math>

and the <math>n \left(\binom{m+r}{r} - 1\right)</math> functions known as the '''derivative coordinates''':

<math display=block>\begin{cases}
  u^\alpha_I:U^k \to \mathbf{R} \\
  u^\alpha_I\left(j^r_p\sigma\right) = \left.\frac{\partial^{|I|} \sigma^\alpha}{\partial x^I}\right|_p
\end{cases}</math>

Given an atlas of adapted charts (''U'', ''u'') on ''E'', the corresponding collection of charts (''U&thinsp;<sup>r</sup>'', ''u&thinsp;<sup>r</sup>'') is a [[finite-dimensional]] ''C''<sup>∞</sup> atlas on ''J&thinsp;<sup>r</sup>''(''π'').