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 $f'(x)\neq 0.$

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