Jet (mathematics)

JETS OF FUNCTIONS BETWEEN EUCLIDEAN SPACES Before giving a rigorous definition of a jet, it is useful to examine some special cases. Example: One-dimensional case Suppose that $f: {\mathbb R} ightarrow{\mathbb R}$ is a real-valued function having at least ''k+1'' Derivatives in a Neighborhood ''U'' of the point $x_0$ . Then by Taylor's theorem, : $f(x)=f(x_0)+f'(x_0)(x-x_0)+\cdots+rac{f^{(k)}(x_0)}{k!}(x-x_0)^{k}+rac{R_{k+1}(x)}{(k+1)!}(x-x_0)^{k+1}$ where
	:<math>J^k P({\mathbb R}^n,{\mathbb R}^m) Q	\left\{J^kf\in J^k_p({\mathbb R}^n,{\mathbb R}^m)f(p)=q ight\}</math>
	::<math>\left Rac{d}{dt} \left( \psi\circ F Ight) (t) Ight {t	0}= \sum_{i=1}^n\leftrac{d}{dt}(x_i\circ f)(t) ight_{t=0}\ (D_i\psi)\circ f(0)</math>

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