Analysis II Lecture 13 Part 1 the differential for functions on manifolds
Channel: Arthur Parzygnat
Date: 2018-05-19

A function on an arbitrary subset of Euclidean space is differentiable at a point c if there exists an open neighborhood of c and an extension of f to that open set that is differentiable. However, defining a differential for that function is more challenging. For manifolds, the tangent space defines a subspace of the tangent space of the Euclidean space at c. It is a fact that the differential of any extension restricts to the tangent space of the manifold in a well-defined manner. This defines the differential of a function that is differentiable at a point on the manifold. Note that there is a tiny abuse of notation as \tilde{f} need not extend all of f---it should just extend f restricted to a small enough neighborhood around c (intersected with the manifold). 

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.