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 Mar13 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. Mar13 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? Mar13 comment Definition of critical point Suppose $df_p=1$. Why is this surjective? Mar13 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"? Mar13 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?! Mar13 asked Definition of critical point Mar5 accepted Definition of operator norm Mar5 asked Definition of operator norm Feb12 revised How to show augmentation ideal is an ideal edited tags Feb12 asked How to show augmentation ideal is an ideal Jan31 accepted Is simple module over commutative ring always a field? Jan31 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... Jan31 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? Jan31 asked Is simple module over commutative ring always a field? Jan30 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? Jan30 asked Non-analytic smooth function Jan30 awarded Yearling Jan30 revised $Rx$ is a simple module deleted 149 characters in body Jan30 accepted $Rx$ is a simple module Jan30 awarded Custodian