Transversal map to a submanifold

Definition

Suppose $M$ is a differential manifold and $N$ is a submanifold. Then, if $P$ is a differential manifold and $f:P\to M$ is a smooth map, we say that $f$ is transversal to $N$ if for any point $x\in N$ and $y\in f^{-1}(n)$, the composite:

$T_{y}P\to ^{Df}T_{x}M\to T_{x}M/T_{x}N$

is surjective. When $N$ is a single point, this is equivalent to saying that that point is a regular value of $f$.

When the map from $P$ to $M$ is also the inclusion of a submanifold, we say that $P$ and $N$ are transversally intersecting submanifolds of $M$.