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 Mar 13 comment Definition of critical point Can I ask what $v$ means in your formula $df_p(v)=f′(p)v$? Mar 13 comment Definition of critical point I think I'm confused about the definition of $df_p$. In my textbook, it seems $df_p$ is the derivative--I'm probably wrong on this, but I have this impression because it can be written as Jacobian matrix. Mar 13 comment Definition of critical point Thanks! Yes, the space spanned by $df_p=1$ is $\mathbb{R}$, but that's not equivalent to $df_p$ being surjective? Mar 13 comment Definition of critical point Suppose $df_p=1$. Why is this surjective? Mar 13 comment Definition of critical point Can I ask the reason why "Since its codomain is 1-dimensional, dfp is either surjective or the zero map"? Mar 13 comment Definition of critical point Sorry that I don't understand your explanation... If at a regular point the differential fails to be surjective, then there'd be something wrong with the definition?! Mar 13 asked Definition of critical point Mar 5 accepted Definition of operator norm Mar 5 asked Definition of operator norm Feb 12 revised How to show augmentation ideal is an ideal edited tags Feb 12 asked How to show augmentation ideal is an ideal Jan 31 accepted Is simple module over commutative ring always a field? Jan 31 comment Is simple module over commutative ring always a field? But after defining this multiplication in $M$, why doesn't this make $M$ a field? Still confused about the logic... Jan 31 comment Is simple module over commutative ring always a field? I'm still confused--why can I treat $M$ as a ring by transferring the structure of $R/I$ if it's not actually a ring? Jan 31 asked Is simple module over commutative ring always a field? Jan 30 comment Non-analytic smooth function @MarianoSuárez-Alvarez If a real analytic function is 0 in an open set in $\mathbb{R}$, can I say it's 0 on the whole real line by this argument? Jan 30 asked Non-analytic smooth function Jan 30 awarded Yearling Jan 30 revised $Rx$ is a simple module deleted 149 characters in body Jan 30 accepted $Rx$ is a simple module