For a smooth map $$f:N\longrightarrow M$$ of manifolds, a point $p$ in $N$ is a critical point of $f$ if the differential $$f_{*,p}:T_{p}N\longrightarrow T_{f(p)}M$$ fails to be surjective.
My problem:
I'm reading some points about map $f:X\longrightarrow Y$ morphism of varieties (Singularities of a Map, Étale maps, degree of maps ...) and I found expressions like:
1) "$x \in X$ a point where the map $f$ is singular."
Question: Is this the same as saying that $ x $ is a critical point of $ f $?
2) "$x \in X$ a point where $df$ is degenerate." $\\$
Quetion: Is this the same as saying that $ x $ is a critical point of $ f $?
Thank You!