I am trying to prepare a presentation on "the use of differential geometry in the theory of critical points", but only the case where there is a single variable. (only in dimension 1), and i ask myself whether the non-degenerate functions make sense in one dimention and how to define it ?
Now I'm working on the book J.Minlor (Morse theory) is it is a little difficult to understand, so if anyone has another book propose to me I'm all ears.
Thank you.
It's simpler than you think:
When $\Omega\subset\Bbb R$ is an open interval and $f:\ \Omega\to\Bbb R$ is a $C^2$ function, a point $a\in\Omega$ is a critical point of $f$ if $f'(a)=0$, and this critical point is nondegenerate if the Hessian of $f$ at $a$ has determinant $\ne0$.
Now what is the Hessian in the one-dimensional case? On a formal level its the $1\times1$-matrix of second partial derivatives of $f$ with respect to all variables, evaluated at $a$. Therefore it is the matrix $\bigl[f''(a)\bigr]$. The determinant of this matrix is the real number $f''(a)$, and if this number is $\ne0$ the critical point $a$ is nondegenerate.