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Uedu Open / Real Analysis
18.100B

Real Analysis

Prof. Tobias Colding | Spring 2025
Science & Math Mathematics Calculus Differential Equations Mathematical Analysis
前往原始課程
CC BY-NC-SA 4.0
課程簡介

This course gives an introduction to analysis, and the goal is twofold:  

         1. To learn how to prove mathematical theorems in analysis and how to write proofs.    
         2. To prove theorems in calculus in a rigorous way.

The course will start with real numbers, limits, convergence, series and continuity.  We will continue on with metric spaces, differentiation and  Riemann integrals.  After that, we will move on to differential equations.

課程資訊
來源MIT 開放式課程
科系Mathematics
語言English
影片數25
課程影片 (25)
1
Lecture 1: Introduction to Real Numbers
Lecture 1: Introduction to Real Numbers
2
Lecture 2: Introduction to Real Numbers (cont.)
Lecture 2: Introduction to Real Numbers (cont.)
3
Lecture 3: How to Write a Proof; Archimedean Property
Lecture 3: How to Write a Proof; Archimedean Property
4
Lecture 4: Sequences; Convergence
Lecture 4: Sequences; Convergence
5
Lecture 5: Monotone Convergence Theorem
Lecture 5: Monotone Convergence Theorem
6
Lecture 6: Cauchy Convergence Theorem
Lecture 6: Cauchy Convergence Theorem
7
Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series
Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series
8
Lecture 8: Convergence Tests for Series; Power Series
Lecture 8: Convergence Tests for Series; Power Series
9
Lecture 9: Limsup and Liminf; Power Series; Continuous Functions; Exponential Function
Lecture 9: Limsup and Liminf; Power Series; Continuous Functions; Exponential Function
10
Lecture 10: Continuous Functions; Exponential Function (cont.)
Lecture 10: Continuous Functions; Exponential Function (cont.)
11
Lecture 11: Extreme and Intermediate Value Theorem; Metric Spaces
Lecture 11: Extreme and Intermediate Value Theorem; Metric Spaces
12
Review for 18.100B Real Analysis Midterm
Review for 18.100B Real Analysis Midterm
13
Lecture 12: Convergence in Metric Spaces; Operations on Sets
Lecture 12: Convergence in Metric Spaces; Operations on Sets
14
Lecture 13: Open and Closed Sets; Coverings; Compactness
Lecture 13: Open and Closed Sets; Coverings; Compactness
15
Lecture 14: Sequential Compactness; Bolzano–Weierstrass Theorem in a Metric Space
Lecture 14: Sequential Compactness; Bolzano–Weierstrass Theorem in a Metric Space
16
Lecture 15: Derivatives; Laws for Differentiation
Lecture 15: Derivatives; Laws for Differentiation
17
Lecture 16: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion
Lecture 16: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion
18
Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals
Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals
19
Lecture 18: Integrable Functions
Lecture 18: Integrable Functions
20
Lecture 19: Fundamental Theorem of Calculus
Lecture 19: Fundamental Theorem of Calculus
21
Lecture 20: Pointwise Convergence; Uniform Convergence
Lecture 20: Pointwise Convergence; Uniform Convergence
22
Lecture 21: Integrals and Derivatives under Uniform Convergence
Lecture 21: Integrals and Derivatives under Uniform Convergence
23
Lecture 22: Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs)
Lecture 22: Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs)
24
Lecture 23: Existence & Uniqueness for ODEs: Picard–Lindelöf Theorem
Lecture 23: Existence & Uniqueness for ODEs: Picard–Lindelöf Theorem
25
Review for the 18.100B Real Analysis Final Exam
Review for the 18.100B Real Analysis Final Exam