Non degeneracy of critical points

[pp31]

In local coordinates what does it mean for a critical point of a function f:M\rightarrowR 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?

 

[mathwonk]

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.