# “Critical point” - single-variable calculus v.s. differential geometry

In differential geometry, given a smooth map $$\Phi: M \rightarrow N$$ between smooth manifolds $$M$$ and $$N$$, a point $$p \in M$$ is said to be a critical point if $$d\Phi_p: T_pM \rightarrow T_{\Phi(p)}N$$ is not surjective.

In single-variable calculus, one learns about differentiating a function and finding its "critical points" for local extrema or inflexions. Does the term "critical point" in single-variable calculus have the same meaning as the differential geometric definition? Or is the terminology just a coincidence?

If it is not the same, what is the intuition behind a "critical point" in the differential geometric definition? I don't really understand it.

For context, I am learning about Sard's Theorem.

In single variable, surjetivity of $$df_x$$ is equivalent to saying that $$df_x\neq 0$$ therefore both definition are equivalent. $$M=N=\mathbb{R}$$ and $$df_x:T_x\mathbb{R}=\mathbb{R}\rightarrow T_{f(x)}\mathbb{R}=\mathbb{R}$$ defined by $$df_x(u)=f'(x)u$$ is a linear function which is surjective if and only if $$f'(x)\neq 0.$$