Definition:Regular Value

Definition
Let $X$ and $Y$ be smooth manifolds.

Let $f: X \to Y$ be a smooth mapping.

Then a point $y \in Y$ is called a regular value of $f$ the pushforward of $f$ at $x$:
 * $f_* \vert_x: T_x X \to T_y Y$

is surjective for every $x \in f^{-1} \left({y}\right) \subseteq X$.

Also defined as
Note that some authors also allow a point $y \in Y$ to be called a regular value, if $f^{-1} \left({y}\right) = \varnothing$.