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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[smooth manifolds|F45VQ47GPV]], let $k \\ge 0$, and let $J^k(X, Y)$ be the jet bundle of $k$-jets of smooth maps from $X$ to $Y$.\nFor a smooth map $f \\colon X \\to Y$, let $j^k f \\colon X \\to J^k(X, Y)$ be its $k$-jet extension, which sends $x$ to the $k$-jet of $f$ at $x$.\nLet $W$ be an [[embedded submanifold|VKFW27WXEZ]] of $J^k(X, Y)$.\nGive $C^\\infty(X, Y)$ the Whitney $C^\\infty$ topology.\nThen the set of maps $f \\in C^\\infty(X, Y)$ such that $j^k f$ is [[transverse|A2PDZ8GA9C]] to $W$ is a [[residual set|GRH8MCVN3N]] in $C^\\infty(X, Y)$, and in particular it is dense.\n\n## Remarks\n\nFor $k = 0$ one has $J^0(X, Y) = X \\times Y$, and with $W = X \\times Z$ the theorem says that maps transverse to a submanifold $Z \\subseteq Y$ are generic.\nThe space $C^\\infty(X, Y)$ with the Whitney topology is a Baire space, so a residual set is dense.\nThe proof reduces to the parametric transversality theorem.\nIf a smooth family $F \\colon X \\times S \\to Y$ is transverse to $Z$, then $F(\\cdot, s)$ is transverse to $Z$ for almost every $s$.\nThis follows from Sard's theorem applied to the projection $F^{-1}(Z) \\to S$.\nLocally one adds polynomial terms of degree at most $k$ to $f$, which moves $j^k f$ in all directions of the fibre.\nExample: let $k = 1$ and let $W$ be the set of $1$-jets whose linear part has rank $r$.\nThe theorem shows that for a generic $f$ the set of points where $df$ has rank $r$ is a submanifold of the expected codimension.\nThis is how Whitney's results on generic singularities are obtained.\n\n## Explanations\n\n### L1 human\n\nA typical smooth map is in general position with respect to any fixed condition on its derivatives.\nWhatever submanifold one fixes in the space of jets, most maps meet it cleanly, and one can always reach such a map by an arbitrarily small change.\nSo in proofs one may assume that the maps one works with are transverse to whatever one likes.\n\n### L2 human\n\nThe theorem is the source of most genericity statements in differential topology and singularity theory.\nWith $k = 0$ it gives transversality of maps to submanifolds, which is used to define intersection numbers and the Pontryagin–Thom construction.\nWith $k = 1$ it shows that generic maps have only mild singularities, for instance that Morse functions are dense and that generic maps $M^n \\to \\R^{2n}$ are immersions.\nWith higher $k$ it is the starting point of the classification of stable singularities.\n",
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