# On the definition of critical point

Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be a smooth function (or in general between two smooth manifolds). Then $p\in \mathbb{R}^n$ is a critical point if $df_p$ is not surjective. I feel confused about this definition. If $n<m$, then $df_p$ can never be surjective, so that every point in $\mathbb{R}^n$ is critical in this case?!

For instance, let $\alpha:\mathbb{R}\to\mathbb{R}^2$ be the curve $\alpha(t)=(\sin t,\cos t)$. Is every $t$ critical?

• Are you suuuuuure that's the definition of critical point? – Robin Goodfellow Mar 13 '15 at 22:09
• @RobinGoodfellow Yes, I'm sure. – Vladimir Mar 13 '15 at 22:23
• @DBS I think Sard says the set of critical values has measure zero, not necessarily the set of critical points. – Ben West Mar 14 '15 at 0:07
• Typically, we define a point to be critical if and only if the pushforward is not injective. This coincides with the standard definition of critical point from elementary calculus (i.e. $f'(t)=0\iff t\text{ is a critical point}$). – Robin Goodfellow Mar 14 '15 at 0:35
• Actually, here's a better link corroborating OP's definition. – André 3000 Mar 14 '15 at 4:02

This may seem not natural in the circle example. There we are dealing with a different matter, that is a regular parametrization of the circle: when you consider that circle $S$ as a manifold (a curve), the mapping $\alpha:\mathbb R\to S$ has all points regular.