Introductory videos

Last year Ulrik Fjordholm made introductory videos to almost all parts of the course. They are highly recommended, and I will link to the relevant ones on the "Before the lectures" page. If you want to look ahead (or look back to review) it may, however, be convenient to have them all listed on the same page, so here they are:

Chapter 1

Section 1.1: Proofs and logic: Video. Notes.

Section 1.2: Sets: Video. Notes.

Section 1.3: Operations with sets: Video. Notes.

Section 1.4: Functions: Video. Notes.

Section 1.6: Cardinality: Video. Notes.

Chapter 2

Section 2.2: Min, max, inf, and sup: Video. Notes.

Section 2.2: liminf and limsup: Video. Notes.

Section 2.2: Completeness of \(\Bbb{R}\) and \(\Bbb{R}^n\): Video. Notes.

Section 2.3: Four theorems from calculus: Video. Notes.

Chapter 3

Section 3.1: Metric spaces: Video. Notes.

Section 3.2: Convergence: Video. Notes.

Section 3.2: Continuity: Video. Notes.

Section 3.3: Open and closed sets: Video. Notes.

Section 3.3: Continuity in terms of open and closed sets: Video. Notes.

Section 3.4: Completeness: Video. Notes.

Section 3.4: Banach's fixed point theorem: Video. Notes.

Section 3.5: Compactness I: Video. Notes.

Section 3.5: Compactness II: Video. Notes.

Section 3.5: Compactness III: Video. Notes.

Section 3.6: Alternative description of compactness: Video. Notes.

Section 3.7: Completions: Video. Notes.

Chapter 4

Section 4.1: Modes of continuity: Video. Notes.

Section 4.2: Modes of convergence: Video. Notes.

Section 4.3: Integrating sequences of functions: Videos. Notes.

Section 4.3: Differentiasting sequences of functions: Videos. Notes.

Section 4.4: Power series: Videos. Notes.

Section 4.5: Spaces of bounded function: Videos. Notes.

Section 4.6: Spaces of continuous functions I: Videos. Notes.

Section 4.7: Ordinary differential equations I: Video. Notes.

Section 4.7: Ordinary differential equations II (uniqueness): Video. Notes.

Section 4.7: Ordinary differential equations III (existence): Video. Notes.

Section 4.8: Arzela-Ascoli's Theorem: Video. Notes.

Section 4.9: Convergence of Euler's method: Video. Notes.

Section 4.10: Weierstrass' approximation theorem I: Video. Notes.

Section 4.10: Weierstrass' approximation theorem II: Video. Notes.

Chapter 5

Chapter 5 background video: Normed spaces: Video. Notes.

Chapter 5 background video: Continuity and convergence in normed spaces: Video. Notes.

Section 5.1: Normed vector spaces I (definitions): Video. Notes.

Section 5.1: Normed vector spaces II (equivalent norms): Video. Notes.

Section 5.2: Series and bases: Video. Notes.

Section 5.3: Inner product spaces: Video. Notes.

Section 5.4: Linear operators I. Video. Notes.

Section 5.4: Linear operators II (boundedness): Video. Notes.

Section 5.4: Linear operators III (spaces of linear operators): Video. Notes.

Section 5.5: Invertible linear operators I: Video. Notes.

Section 5.5: Invertible linear operators II (Neumann series): Video. Notes.

Chapter 6

Chapter 6 background video: What is a derivative: Video. Notes.

Section 6.1: The Frechet derivative I: Video. Notes.

Section 6.1: The Frechet derivative II: Video. Notes.

Section 6.2: The Gateaux derivative: Video. Notes.

Section 6.3: The mean value theorem: Video. Notes.

Section 6.5: Multiindices: Video. Notes.

Section 6.5: Taylor's formula (in \(\Bbb{R}^d\)): Video. Notes.

Section 6.6: Partial derivatives: VideoNotes.

Section 6.7: The inverse function theorem: Video. Notes.

Section 6.8: The implicit function theorem I (motivation): Video. Notes.

Section 6.8: The implicit function theorem II (proof): Video. Notes.

Chapter 10

Section 10.1: Fourier series (introduction and motivation): Video. Notes.

Section 10.2: Convergence of Fourier series I: Video. Notes.

Section 10.2: Convergence of Fourier series II: Video. Notes.

Section 10.3: The Dirichlet kernel: Video. Notes.

Section 10.4: Cesaro convergence: Video. Notes.

Publisert 12. jan. 2022 13:56 - Sist endret 2. des. 2023 07:10