Overview

This course covers topics in Analysis II, including functions, limits, derivatives, integrals, manifolds, and differential equations. Students will learn about basic topology, continuity, differentiability, vector fields, and theorems related to manifolds. The course also delves into metric spaces, point set topology, and the method of successive approximations. The intended audience for this course is learners interested in advanced mathematical concepts and applications.

Syllabus

Introduction to LaTeX and TikZ.
Analysis II Lecture 01 Part 1 diagrams.
Analysis II Lecture 01 Part 2 products.
Analysis II Lecture 01 Part 3 existence and uniqueness of products.
Analysis II Lecture 01 Part 4 determinants.
Analysis II Lecture 02 Part 1 basic topology of euclidean space.
Analysis II Lecture 02 Part 2 nested rectangles.
Analysis II Lecture 02 Part 3 Compactness.
Analysis II Lecture 02 Part 4 connected and convex subsets.
Analysis II Lecture 03 Part 1 functions.
Analysis II Lecture 03 Part 2 limits.
Analysis II Lecture 03 Part 3 continuity.
Analysis II Lecture 03 Part 4 continuity theorems.
Analysis II Lecture 03 Part 5 continuous paths.
Analysis II Lecture 04 Part 1 intuition for derivatives.
Analysis II Lecture 04 Part 2 the differential.
Analysis II Lecture 04 Part 3 the chain rule.
Analysis II Lecture 04 Part 4 example applying the chain rule.
Analysis II Lecture 05 Part 1 partial derivatives.
Analysis II Lecture 05 Part 2 continuously differentiable functions.
Analysis II Lecture 06 Part 1 The derivative functor.
Analysis II Lecture 06 Part 2 vector fields as derivations.
Analysis II Lecture 06 Part 3 when partial derivatives commute.
Analysis II Lecture 06 Part 4 continuously differentiable versus differentiable.
Analysis II Lecture 07 Part 1 integral curves of vector fields.
Analysis II Lecture 07 Part 2 dynamical systems.
Analysis II Lecture 07 Part 3 integrals/constants of the motion.
Analysis II Lecture 08 Part 1 inverse differential.
Analysis II Lecture 08 Part 2 motivation for the inverse function theorem.
Analysis II Lecture 08 Part 3 sketch of proof of inverse function theorem I.
Analysis II Lecture 08 Part 4 sketch of proof of inverse function theorem II.
Analysis II Lecture 09 Part 1 (review) example computing the differential of a function.
Analysis II Lecture 10 Part 1 height functions and level sets.
Analysis II Lecture 10 Part 2 Lemma for the implicit function theorem.
Analysis II Lecture 10 Part 3 proof of lemma for the implicit function theorem.
Analysis II Lecture 10 Part 4 statement and example of implicit function theorem.
Analysis II Lecture 11 Part 1 manifolds.
Analysis II Lecture 11 Part 2 alternative definition of manifold and non-examples.
Analysis II Lecture 11 Part 3 implicitly defined manifolds.
Analysis II Lecture 12 Part 1 the tangent space.
Analysis II Lecture 12 Part 2 tangent space using curves.
Analysis II Lecture 12 Part 3 associative algebras and derivations.
Analysis II Lecture 12 Part 4 Hadamard's Lemma.
Analysis II Lecture 13 Part 1 the differential for functions on manifolds.
Analysis II Lecture 13 Part 2 Jacobians for differentiable functions on manifold.
Analysis II Lecture 13 Part 3 familiar theorems for manifolds.
Analysis II Lecture 13 Part 4 submanifolds and normal vectors.
Analysis II Lecture 14 Part 1 orientations.
Analysis II Lecture 14 Part 2 the degree and index.
Analysis II Lecture 14 Part 3 examples of the index for vector fields.
Analysis II Lecture 14 Part 4 the index is well-defined.
Analysis II Lecture 15 Part 1 vector fields on manifolds.
Analysis II Lecture 15 Part 2 flows on manifolds.
Analysis II Lecture 15 Part 3 Triangulations and the Euler characteristic.
Analysis II Lecture 15 Part 4 Poincare Hopf theorem and hairy ball theorem.
Analysis II Lecture 16 Part 1 metric spaces.
Analysis II Lecture 16 Part 2 Cauchy sequences in metric spaces.
Analysis II Lecture 16 Part 3 point set topology and types of functions.
Analysis II Lecture 16 Part 4 the completion of a metric space.
Analysis II Lecture 17 Part 1 the method of successive approximations.
Analysis II Lecture 17 Part 2 contraction mapping theorem I.
Analysis II Lecture 17 Part 3 contraction mapping theorem II.
Analysis II Lecture 17 Part 4 weaker fixed point theorem for compact subsets.
Analysis II Lecture 18 Part 1 the matrix exponential.
Analysis II Lecture 18 Part 2 damped harmonic oscillator.
Analysis II Lecture 18 Part 3 non-autonomous linear ordinary differential equations.
Analysis II Lecture 19 Part 1 integral equations.
Analysis II Lecture 19 Part 2 existence and uniqueness of solutions to ODEs.