Need reference on indicies of critical points I came across the term "index-1 critical point" in my reading, and I would like to know if there are some good references to learn about indices of critical points of smooth functions.
The wikipedia page just defines what they are http://en.wikipedia.org/wiki/Critical_point_(mathematics), but I like to know more about them. 
Unfortunately, I have no background in differential geometry at all,so any reference that does not require substantial background in differential geometery would be greatly appreciated! :) 
 A: Depending what you want to know, there's not much to say. Let's assume $p$ is a non-degenerate critical point (i.e., the Hessian $H(p)$ is non-singular) of a function $f$ having continuous second partial derivatives. By equality of mixed partial derivatives, the Hessian is a symmetric matrix, and therefore admits an orthonormal basis of eigenvectors. The index of the critical point $p$ is the sum of the dimensions of the eigenspaces with negative eigenvalue. Loosely this is "the number of directions you can approach $p$ from below" (i.e., through values of $f$ smaller than $f(p)$).
The classic example is a torus resting on edge on a flat table top, with $f$ the height about the table. There are four critical points: the very bottom (a local minimum, so of index $0$); the bottom and top of the hole (each of index $1$; there's a unique eigendirection through which you can "ascend" into each critical point); and the very top (an absolute maximum for height, and therefore of index $2$ since the torus is $2$-dimensional).
If you need more than the linked Wikipedia page on Morse theory, the Standard Reference is probably Milnor's Morse Theory.
