Jet space of a scalar ordinary differential equation

The $n$th jet space for a scalar function $u(x)$ has local coordinatesIt records the value of a function and its derivatives through order $n$ at one point, allowing an order-$n$ ordinary differential equation to be treated as a hypersurface and a point transformation to act through its prolongation of a vector field.

An ordinary differential equation $\Delta (x,u,u^{\prime },\dots ,u^{(n)})=0$ defines a solution manifold in jet space. The prolonged graph of every classical solution lies in this manifold.