Analysis II Lecture 13 Part 2 Jacobians for differentiable functions on manifold
Channel: Arthur Parzygnat
Date: 2018-05-19

We review some linear algebra by describing what it means to choose a basis on a finite-dimensional vector space and what it means to write a matrix of a linear transformation with respect to an ordered basis. This can be used to define the Jacobian of a differentiable function at any point on the tangent space by choosing bases for the tangent spaces at the source and the target. Categorically, choosing such bases on vector spaces and matrices for linear transformations corresponds to choosing an adjoint equivalence between the category of Euclidean spaces with matrices and the category of finite-dimensional vector spaces with linear transformations.

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.