= Jet bundle of maps
{title2=$J^k(M,N)$}
The space of Taylor-equivalence classes $j_x^kf$ of smooth maps $M^m\to N^n$, with derivatives through order $k$ agreeing at the source point. The projection $j_x^kf\mapsto(x,f(x))$ has total dimension $m+n\binom{m+k}{k}$ and fiber dimension $n(\binom{m+k}{k}-1)$. First-order fibers are canonically $\operatorname{Hom}(T_xM,T_yN)$; higher-order fibers have coordinate-dependent Taylor descriptions. The truncation to order $k-1$ is an <affine bundle> modeled on $\operatorname{Sym}^k(T_x^*M)\otimes T_yN$.
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