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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[smooth manifolds|F45VQ47GPV]], let $Z \\subseteq Y$ be an [[embedded submanifold|VKFW27WXEZ]], and let $f \\colon X \\to Y$ be a smooth map.\nThe map $f$ is *transverse* to $Z$, written $f \\pitchfork Z$, if for every $x \\in X$ with $f(x) \\in Z$,\n$$df_x(T_xX) + T_{f(x)}Z = T_{f(x)}Y,$$\nwhere $df_x$ is the [[differential|6P9PXQDSEN]] of $f$ at $x$.\nTwo embedded submanifolds $A, B \\subseteq Y$ are *transverse*, written $A \\pitchfork B$, if $T_pA + T_pB = T_pY$ for every $p \\in A \\cap B$; this is transversality of the inclusion $A \\to Y$ to $B$.\n\n## Remarks\n\nThe condition is empty where $f(X)$ misses $Z$, so a map whose image avoids $Z$ is transverse to $Z$.\nIf $Z$ is a point $y$, then $f \\pitchfork \\set{y}$ means that $y$ is a regular value of $f$.\nIf $f \\pitchfork Z$, then $f^{-1}(Z)$ is an embedded submanifold of $X$ with $\\dim X - \\dim f^{-1}(Z) = \\dim Y - \\dim Z$.\nExample: in $\\R^2$ the $x$-axis and the line $y = x$ are transverse at the origin, since their directions $(1, 0)$ and $(1, 1)$ span $\\R^2$.\nNon-example: the $x$-axis and the parabola $y = x^2$ meet at the origin, where both tangent lines equal the $x$-axis, which does not span $\\R^2$; so they are not transverse.\n",
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