Analysis II Lecture 14 Part 1 orientations
Channel: Arthur Parzygnat
Date: 19/05/2018

An orientation on a vector space is a choice of an equivalence class of ordered bases where two ordered bases are equivalent iff the linear transformation sending one to the other has positive determinant. An orientation on a manifold is a choice of orientations on each of the tangent spaces such that there exists a chart that preserves the orientation from Euclidean space.

Note: When making the claim that orientable manifolds have two orientations, I forgot to say "connected." The correct statement is "Every connected component of an orientable manifold has two orientations." The following definition for connectedness suffices: "A manifold M is connected iff given any two points x,y in M there exists a smooth path whose starting point is x and ending point is y" (or see https://youtu.be/iLWbAX_kC44 for the aprior more general definition). Thanks to Caleb Nastasi for catching my typo at the beginning and end of the video.

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.