Analysis II Lecture 14 Part 2 the degree and index
Channel: Arthur Parzygnat
Date: 19/05/2018

The degree of a differentiable function from a compact oriented manifold to a connected oriented manifold of the same dimension is the sum of the signs of the differentials associated to any regular value. A special case of the degree can be defined given a vector field on Euclidean space at the critical point of the vector field by normalizing the vector field on a small enough sphere centered at that critical point.

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.