Cartesian jet of a function

For a class $C^k$ function $f:U\subseteq\mathbb R^n\to\mathbb R^m$ , its Cartesian $k$ -jet at $a\in U$ is the array

j_a^k f=(\partial^\alpha f(a))_{|\alpha|\le k}.

Here $\alpha$ is a multi-index. Two functions have the same $k$ -jet when all these derivative values agree. The same data can be encoded by the polynomial

\sum_{|\alpha|\le k}\frac{\partial^\alpha f(a)}{\alpha!}(x-a)^\alpha.

Infinite jets and coordinate choices

An infinite jet specifies these values for every order; it does not assert convergence of the associated formal Taylor series. Cartesian jets use fixed Euclidean coordinates. The jet of a smooth map is the corresponding coordinate-independent geometric construction.