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 $f$ if and only if the pushforward of $f$ at $x$:

$f_* \vert_x: T_x X \to T_y Y$

This article, or a section of it, needs explaining. In particular: What do all the symbols mean in this context? Presume $\vert$ might mean restriction, but this is not obvious (if so then use $\restriction$); if "pushforward" actually means $f_* \vert_x: T_x X \to T_y Y$ then set up the page to define it, thus doing all the hard work of defining that concept all in one place. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

is surjective for every $x \in \map {f^{-1} } y \subseteq X$.

Also defined as

Note that some authors also allow a point $y \in Y$ to be called a regular value, if $\map {f^{-1} } y = \O$.