Editing Jet (mathematics) (section)

===One-dimensional case===

Suppose that <math>f: {\mathbb R}\rightarrow{\mathbb R}</math> is a real-valued function having at least ''k''&nbsp;+&nbsp;1 [[derivative]]s in a [[neighbourhood (mathematics)|neighborhood]] ''U'' of the point <math>x_0</math>. Then by Taylor's theorem,

:<math>f(x)=f(x_0)+f'(x_0)(x-x_0)+\cdots+\frac{f^{(k)}(x_0)}{k!}(x-x_0)^{k}+\frac{R_{k+1}(x)}{(k+1)!}(x-x_0)^{k+1}</math>

where
:<math>|R_{k+1}(x)|\le\sup_{x\in U} |f^{(k+1)}(x)|.</math>
Then the '''''k''-jet''' of ''f'' at the point <math>x_0</math> is defined to be the polynomial
:<math>(J^k_{x_0}f)(z)
=\sum_{i=0}^k \frac{f^{(i)}(x_0)}{i!}z^i
=f(x_0)+f'(x_0)z+\cdots+\frac{f^{(k)}(x_0)}{k!}z^k.</math>

Jets are normally regarded as [[Polynomial#Abstract algebra|abstract polynomials]] in the variable ''z'', not as actual polynomial functions in that variable. In other words, ''z'' is an [[indeterminate (variable)|indeterminate variable]] allowing one to perform various [[abstract algebra|algebraic operations]] among the jets. It is in fact the base-point <math>x_0</math> from which jets derive their functional dependency. Thus, by varying the base-point, a jet yields a polynomial of order at most ''k'' at every point. This marks an important conceptual distinction between jets and truncated [[Taylor series]]: ordinarily a Taylor series is regarded as depending functionally on its variable, rather than its base-point. Jets, on the other hand, separate the algebraic properties of Taylor series from their functional properties. We shall deal with the reasons and applications of this separation later in the article.