Analysis II Lecture 11 Part 3 implicitly defined manifolds
Channel: Arthur Parzygnat
Date: 2018-05-19

Implicitly defined manifolds are manifolds that are defined as level sets of functions. The critical points, regular values, and regular points of a differentiable function are defined. There is a theorem that says the inverse image of a regular value for a differentiable function is a differentiable manifold. Note: there is a mistake in the calculation of the regular points of f in the example given. It should say R^2 minus all the points along the lines y=x and y=-x since those points are in the inverse image of 0 along f.

This is part of a series of lectures on Mathematical Analysis II. Topics covered include continuous and differentiable multi-variable functions on Euclidean space, the chain rule, the implicit function theorem, manifolds, tangent spaces, vector fields, the degree and index of a smooth map, the Euler characteristic, metric spaces, the contraction mapping theorem, existence and uniqueness of solutions to ordinary differential equations, and integral equations.

I speak rather slowly, so you may wish to increase the speed of this video.

These videos were created during the 2017 Spring semester at the UConn CETL Lightboard Room.