Analysis II Lecture 10 Part 2 Lemma for the implicit function theorem

This is a rather technical lemma related to the implicit function theorem. I have tried to express it more diagrammatically, which is why it might look more complicated than what you would find in most references. This is my personal preference because the proof drastically simplifies from the perspective. This video will make much more sense when you see the proof of the claim so I highly recommend watching Lecture 10 Part 3 immediately after this. 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.