Jet bundle of maps ($J^k(M,N)$)

The space of Taylor-equivalence classes $j_x^{k}f$ of smooth maps $M^m\to N^n$, with derivatives through order $k$ agreeing at the source point. The projection $j_x^{k}f\mapsto (x,f(x))$ has total dimension $m+n\left(\genfrac{}{}{0ex}{}{m+k}{k}\right)$ and fiber dimension $n((\genfrac{}{}{0ex}{}{m+k}{k})-1)$. First-order fibers are canonically $\mathrm{Hom}(T_{x}M,T_{y}N)$; higher-order fibers have coordinate-dependent Taylor descriptions. The truncation to order $k-1$ is an affine bundle modeled on ${\mathrm{Sym}}^k(T_x^{\ast }M)\otimes T_{y}N$.