Non degeneracy of critical points

by pp31
Tags: critical, degeneracy, points
pp31 is offline
Nov26-10, 09:22 PM
P: 10
In local coordinates what does it mean for a critical point of a function f:M[tex]\rightarrow[/tex]R to be non degenerate?
In addition how can you show that the definition is independent of the choice of
local coordinates?

I know that being a critical point is independent of the choice of local coordinates but
I am struggling with the second derivate in local coordinates.
Any help is appreciated?
Phys.Org News Partner Science news on
Going nuts? Turkey looks to pistachios to heat new eco-city
Space-tested fluid flow concept advances infectious disease diagnoses
SpaceX launches supplies to space station (Update)
mathwonk is offline
Nov28-10, 12:02 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,421
well at a critical point, all 1st derivatives are zero, and then you have as next best thing, the symmetric "Hessian" matrix of 2nd derivatives. non degenerate means that Hessian matrix is invertible. Such a point is also called an "ordinary double point". The most naive way to check that is an invariant notion is to go in there and slog it out with a coordinate change by the chain rule.

Register to reply

Related Discussions
Critical Points of f(x,y) Calculus & Beyond Homework 1
Critical points Calculus 2
Critical Points General Math 7
Critical Points Calculus & Beyond Homework 1
critical points Introductory Physics Homework 1