Editing Jet bundle (section)

===Example===
If π is the [[trivial bundle]] (''M'' × '''R''', pr<sub>1</sub>, ''M''), then there is a canonical [[diffeomorphism]] between the first jet bundle <math>J^1(\pi)</math> and ''T*M'' × '''R'''. To construct this diffeomorphism, for each σ in <math>\Gamma_M(\pi)</math> write <math>\bar{\sigma} = pr_2 \circ \sigma \in C^\infty(M)\,</math>.

Then, whenever ''p'' ∈ ''M''

:<math>j^1_p \sigma = \left\{ \psi : \psi \in \Gamma_p (\pi); \bar{\psi}(p) = \bar{\sigma}(p); d\bar{\psi}_p = d\bar{\sigma}_p \right\}. \,</math>

Consequently, the mapping

:<math>\begin{cases}
  J^1(\pi) \to T^*M \times \mathbf{R} \\
  j^1_p\sigma \mapsto \left(d\bar{\sigma}_p, \bar{\sigma}(p)\right)
\end{cases}</math>

is well-defined and is clearly [[injective]]. Writing it out in coordinates shows that it is a diffeomorphism, because if ''(x<sup>i</sup>, u)'' are coordinates on ''M'' × '''R''', where ''u'' = id<sub>'''R'''</sub> is the identity coordinate, then the derivative coordinates ''u<sub>i</sub>'' on ''J<sup>1</sup>(π)'' correspond to the coordinates ∂<sub>''i''</sub> on ''T*M''.

Likewise, if π is the trivial bundle ('''R''' × ''M'', pr<sub>1</sub>, '''R'''), then there exists a canonical diffeomorphism between <math>J^1(\pi)</math>and '''R''' × ''TM''.