Editing Jet bundle (section)

==Partial differential equations==
Let ''(E, π, M)'' be a fiber bundle. An '''''r''-th order [[partial differential equation]]''' on π is a [[closed manifold|closed]] [[embedding|embedded]] submanifold ''S'' of the jet manifold ''J<sup>r</sup>(π)''. A solution is a local section σ ∈ Γ<sub>''W''</sub>(π) satisfying <math>j^{r}_p\sigma \in S</math>, for all ''p'' in ''M''.

Consider an example of a first order partial differential equation.

===Example===
Let π be the trivial bundle ('''R'''<sup>2</sup> × '''R''', pr<sub>1</sub>, '''R'''<sup>2</sup>) with global coordinates (''x''<sup>1</sup>, ''x''<sup>2</sup>, ''u''<sup>1</sup>). Then the map ''F'' : ''J''<sup>1</sup>(π) → '''R''' defined by

:<math>F = u^1_1 u^1_2 - 2x^2 u^1</math>

gives rise to the differential equation

:<math>S = \left\{j^1_p\sigma \in J^1\pi\ :\ \left(u^1_1u^1_2 - 2x^2u^1\right)\left(j^1_p\sigma\right) = 0\right\}</math>

which can be written

:<math>\frac{\partial \sigma}{\partial x^1}\frac{\partial \sigma}{\partial x^2} - 2x^2\sigma = 0.</math>

The particular

:<math>\begin{cases}
  \sigma : \mathbf{R}^2 \to \mathbf{R}^2 \times \mathbf{R} \\
  \sigma(p_1, p_2) = \left( p^1, p^2, p^1(p^2)^2 \right)
\end{cases}</math>

has first prolongation given by

:<math>j^1\sigma\left(p_1, p_2\right) = \left( p^1, p^2, p^1\left(p^2\right)^2, \left(p^2\right)^2, 2p^1 p^2 \right) </math>

and is a solution of this differential equation, because

:<math>\begin{align}
  \left(u^1_1 u^1_2 - 2x^2 u^1 \right)\left(j^1_p\sigma\right)
    &= u^1_1\left(j^1_p\sigma\right)u^1_2\left(j^1_p\sigma\right) - 2x^2\left(j^1_p\sigma\right)u^1\left(j^1_p\sigma\right) \\
    &= \left(p^2\right)^2 \cdot 2p^1 p^2 - 2 \cdot p^2 \cdot p^1\left(p^2\right)^2 \\
    &= 2p^1\left(p^2\right)^3 - 2p^1 \left(p^2\right)^3 \\
    &= 0
\end{align}</math>

and so <math>j^1_p\sigma \in S</math> for ''every'' ''p'' ∈ '''R'''<sup>2</sup>.