Non degeneracy of critical points

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SUMMARY

A critical point of a function f:M→R is defined as non-degenerate when the Hessian matrix of second derivatives is invertible. This property is independent of the choice of local coordinates, as critical points remain consistent across transformations. To demonstrate this independence, one can apply the chain rule to analyze the behavior of the Hessian under coordinate changes. Non-degenerate critical points are also referred to as "ordinary double points."

PREREQUISITES
  • Understanding of critical points in differential geometry
  • Familiarity with Hessian matrices and their properties
  • Knowledge of local coordinates and coordinate transformations
  • Proficiency in applying the chain rule in calculus
NEXT STEPS
  • Study the properties of Hessian matrices in detail
  • Learn about coordinate transformations in differential geometry
  • Explore the implications of non-degenerate critical points in optimization
  • Investigate the relationship between critical points and stability in dynamical systems
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Mathematicians, physicists, and students in advanced calculus or differential geometry who are analyzing critical points and their properties in various contexts.

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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?
 
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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.
 

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