Differential geometry MOC

Critical points/values and regular points/values

Let be a differentiable map between differentiable manifolds of dimensions respectively. A point is called a critical point iff where is the Tangent map at ; otherwise is a regular point (and is submersive at ). The image of a critical point is a critical value, and a value which is not critical is a regular value.

Special cases

Single variable function

A critical point is either a Local extremum or a point of inflection (POI). It is defined as a point where the derivative is either 0 or undefined. #to/generalize

Classifying critical points

First derivative test

Given ,

  1. Iff. and then is a Local maximum.
  2. Iff. and then is a Local minimum.
  3. Otherwise, it is neither.
Second derivative test

We can also use the second derivative to test for Concavity at a critical point, which can classify a critical point. Given ,

  1. Iff. the critical point is Concave down and is a Local maximum.
  2. Iff. the critical point is Concave up and is a Local minimum.

#state/tidy | #SemBr