我的数学学习之路，基于MIT Roadmaps，教材选择参考：
- Math.stackexchange
- cargalmathbooks
- https://freecomputerbooks.com/puremath.htm
- 常用教科書列表與簡評 | 臺大數學系 (ntu.edu.tw)
每个内容学习两遍，一遍初学，一遍Review，Review结束之后解决掉所有的Todo。将所有数学基础知识学习贯通，准备好进行数学研究。
- 基础微积分
- 数学推理入门（Logic, mathematical reasoning/inference/proof）
- Book of proof 阅读并完成所有习题：Book of proof Todo
- 离散数学：离散数学书籍列表
- 不等式相关（需要强化训练）：不等式书籍列表
- 数列及其放缩技巧学习
- 初等数论（需要强化训练）
- 《初等数论及其应用》Rosen 阅读，目标是掌握一些基本的定理和证明，为后期的数论学习和证明方法学习做铺垫，以免再学习证明的过程中因为数论的知识缺憾而无法下手解题
- 数学归纳法（需要强化训练）
- 初等数学复习
- 数理化自学全书
- 高中数理化甲种本
- 中学实验课本
- 奥赛小蓝皮
- 解题方法论
- Polya 如何解题
- How to Prove it, by Daniel Velleman
- Terrence Tao - Solving mathematical problems
- How to think about analysis
- Introduction to Mathematical Thinking
- 数学与猜想
- 数学的发现
- 数学证明
- Amazon.com: Problem-Solving Strategies (Problem Books in Mathematics): 9780387982199: Engel, Arthur: Books
- 数学推理复习巩固（需要强化训练）
- 首先去学习初等数论和组合学，学完以后再来复习
proof的相关知识，否则会因为基础不好而受到打击，本部分学到对于任意的proof都能产生敏感的直觉为止 - 搜集好的关于数学推理和证明的教材
- How to prove it? A Structured approach 阅读
- An Introduction to Mathematical Reasoning: Numbers, Sets and Functions : Eccles, Peter J. (University of Manchester)
- Induction and Analogy in Mathematics
- A Transition to Mathematics with Proofs (International Series in Mathematics): 9781449627782: Cullinane, Michael J: Books
- Journey into Mathematics: An Introduction to Proofs (Dover Books on Mathematics): Rotman, Joseph J.: 9780486453064: Amazon.com: Books
- Proofs and Fundamentals: A First Course in Abstract Mathematics: 0 : Bloch, Ethan D.: Amazon.sg: Books
- Proof in Mathematics: An Introduction: Franklin, James, Daoud, Albert: 9780646545097: Amazon.com: Books
- Mathematical Proofs: A Transition to Advanced Mathematics : Chartrand, Gary, Polimeni, Albert D., Zhang, Ping: Amazon.sg: Books
- 实现一个软件，可以把英文转化为数学公式，推理数学公式，也可以证明数学定理
- 首先去学习初等数论和组合学，学完以后再来复习
- 数学思想 & 数学史
- 数学分析
- 高等代数
- 常微分方程
- 概率论与数理统计
- 抽象代数
- 代数学
- 拓扑学
- 数理逻辑
- 组合数学
- 布鲁迪《组合数学》
- 高纳德《具体数学》
- 图论
纯数学习路径完成
- 18.02 Multivariable Calculus
- The Calculus: A Genetic Approach: Toeplitz, Otto, Bressoud, David: 9780226806686: Amazon.com: Books
- Calculus Made Easy by Silvanus P. Thompson
- Calculus: One-variable calculus, with an introduction to linear algebra
- Calculus: Multi-variable calculus and linear algebra, with applications to differential equations and probability
- 18.06 Linear Algebra
- Gilbert Strang <Introduction to Linear Algebra> 阅读完成，并完成所有习题
- Chapter 1-2 阅读并完成习题
- 2.7 习题完成
- Chapter 3 习题完成
- Chapter 4 阅读完成
- Lecture 15 子空间投影观看
- 4.2 Projections 阅读
- 4.2 习题完成
- Lecture 16 投影矩阵，最小二乘观看
- 4.3 Least Square Approximations 阅读
- 4.3 习题完成
- Lecture 17 正交矩阵，Schmidt 正交化观看 (QR 分解还不是很理解)
- 4.4 Orthonormal Bases and Gram-Schmidt 阅读
- 4.4 习题完成
- Chapter 5 阅读完成
- Lecture 18 行列式及其性质
- 5.1 The Properties of Determinants 阅读
- 5.1 习题完成
- Lecture 19 行列式公式，代数余子式
- 5.2 Permutations and Cofactors 阅读
- 5.2 习题完成
- Lecture 20 克拉默法则，逆矩阵，体积
- 5.3 Cramer's Rule, Inverses, and Volumes 阅读
- 5.3 习题完成
- Chapter 6 阅读完成
- Lecture 21 特征值，特征向量阅读
- 6.1 Introduction to Eigenvalues 阅读
- 6.1 习题完成
- Lecture 22 对角化，A 的幂阅读
- 6.2 Diagonalizing a Matrix 阅读
- 6.2 习题完成
- Lecture 23 微分方程，$\exp(At)$ 阅读
从这一节开始必须学习 MIT 的微分方程课程，不然听不懂。
- 6.3 Systems of Differential Equations 阅读
- 6.3 习题完成
- 10.1 阅读
- 10.1 习题完成
- 10.2 阅读
- 10.2 习题完成
- Lecture 24 马尔科夫矩阵，傅里叶级数
- 10.3 Markov Matrices, Population, and Economics 阅读
- 10.3 习题完成
- 10.4 阅读
- 10.4 习题完成
- 10.5 Fourier Series: Linear Algebra for Functions 阅读
- 10.5 习题完成
- Lecture 25 复习二
- Lecture 26 对称矩阵及正定性
- 6.4 Symmetric Matrices 阅读
- 6.4 习题完成
- Lecture 27 复数矩阵，快速傅里叶变换
- Chapter 9 Complex Vectors and Matrices 阅读
- Chapter 9 习题完成
- Lecture 28 正定矩阵，最小值
- 6.5 Positive Definite Matrices 阅读
- 6.5 习题完成
- Lecture 29 相似矩阵，若尔当型
- Chapter 8 阅读
- Chapter 8 习题完成
- Chpater 7 阅读完成
- Lecture 30 奇异值分解
- Chapter 7 The Singular Value Decomposition (SVD) 阅读
- Chapter 7 习题完成
- Lecture 31 线性变换及对应矩阵
- Chapter 8 Linear Transformations 阅读
- Chapter 8 习题完成
- Lecture 32 基变换，图像压缩
- Lecture 33 单元检测 3 复习
- Lecture 34 左右逆，伪逆
- 7.4 The Geometry of the SYD 阅读
- 7.4 习题完成
- Lecture 35 期末复习
- Chapter 8 阅读完成
- Chapter 9 阅读完成
- Chapter 10 阅读完成
- Chapter 11 阅读完成
- Chapter 12 阅读完成
- 全书总结完成
- 18.06 课程资料学习
- 《线性代数及其应用》阅读完成，并完成所有习题
- 其他线代课本阅读并完成习
- Gilbert Strang <Introduction to Linear Algebra> 阅读完成，并完成所有习题
- 18.03 x Differential Equations
- William A. Adkins • Mark G. Davidson 《Ordinary Differential Equations》阅读
- 18.03
- Part I - First-order differential equations
- R 1 Natural growth, separable equations
- R 1 Reading
- L 1 Direction fields, existence and uniqueness of solutions
- L 1 Reading
- PS 1
- R 2 Direction fields, integral curves, isoclines, separatrices, funnels
- R 2 Reading
- L 2 Numerical Methods
- L 2 Reading
- L 3 Linear equations, models
- L 3 Reading
- R 3 Euler’s method; linear Models
- R 3 Reading
- L 4 Solution of linear equations, integrating factors
- L 4 Reading
- R 4 First order linear ODEs; integrating factors
- R 4 Reading
- L 5 Complex numbers, roots of unity
- L 5 Reading
- PS 1 Due, PS 2 Out
- L 6 Complex exponentials; sinusoidal functions
- L 6 Reading
- L 7 Linear system response to exponential and sinusoidal input; gain, phase lag
- L 7 Reading
- R 5 Complex numbers; complex exponentials
- R 5 Reading
- L 8 Autonomous equations; the phase line, stability
- L 8 Reading
- PS 2 due; PS 3 out
- L 9 Linear vs. Nonlinear
- L 9 Reading
- R 6 Review for exam I
- R 6 Reading
- Exam 1
- Part II - Second-order linear equations
- R 7 Solutions to second order ODEs + Readings
- L 11 Modes and the characteristic polynomial + Readings
- L 12 Good vibrations, damping conditions + Readings
- R 8 Homogeneous 2nd order linear constant coefficient equations + Readings
- L 13 Exponential response formula, spring drive + Readings
- R 9 Exponential and sinusoidal input signals + Readings
- L 14 Complex gain, dashpot drive + Readings
- PS 3 due; PS 4 out
- L 15 Operators, undetermined coefficients, resonance + Readings
- R 10 Gain and phase lag; resonance; undetermined coefficients + Readings
- L 16 Frequency response + Readings
- R 11 Frequency response + Readings
- L 17 LTI systems, superposition, RLC circuits. + Readings
- PS4 due; PS 5 out
- L 18 Engineering applications + Readings
- R 12 Review for exam II + Readings
- L 19 Exam II + Readings
- Hour Exam II
- Part III. Fourier series
- R 13 Fourier series: introduction + Readings
- L 20 Fourier series + Readings
- L 21 Operations on fourier series + Readings
- R 14 Fourier series + Readings
- L 22 Periodic solutions; resonance + Readings
- R 15 Fourier series: harmonic response + Readings
- L 23 Step functions and delta functions + Readings
- PS 5 due; PS 6 out
- L 24 Step response, impulse response + Readings
- R 16 Step and delta functions, and step and delta responses + Readings
- L 25 Convolution + Readings
- R 17 Convolution + Readings
- L 26 Laplace transform: basic properties + Readings
- PS 6 due; PS 7 out
- L 27 Application to ODEs + Readings
- R 18 Laplace transform + Readings
- L 28 Second order equations; completing the squares + Readings
- R 19 Laplace transform II + Readings
- L 29 The pole diagram + Readings
- PS 7 due; PS 8 out
- L 30 The transfer function and frequency response + Readings
- R 20 Review for exam III + Readings
- Hour Exam III
- Part IV. First order systems
- L 32 Linear systems and matrices + Readings
- R 21 First order linear systems + Readings
- L 33 Eigenvalues, eigenvectors + Readings
- R 22 Eigenvalues and eigenvectors + Readings
- L 34 Complex or repeated eigenvalues + Readings
- PS 8 due; PS 9 out
- L 35 Qualitative behavior of linear systems; phase plane + Readings
- R 23 Linear phase portraits + Readings
- L 36 Normal modes and the matrix exponential + Readings
- R 24 Matrix exponentials + Readings
- L 37 Nonlinear systems + Readings
- PS 9 due
- L 38 Linearization near equilibria; the nonlinear pendulum + Readings
- R 25 Autonomous systems + Readings
- L 39 Limitations of the linear: limit cycles and chaos + Readings
- R 26 Reviews + Readings
- Final Exam
- Part I - First-order differential equations
- 18.03 SC
- 18.034 Honor Differential Equations
- Elementary Differential Equations with Boundary Value Problems 阅读
- Chapter 1 First-Order Differential Equations 阅读
- Chapter 2 Linear Equations of Higher Order 阅读
- Chapter 3 Power Series Methods 阅读
- Chapter 4 Laplace Transform Methods 阅读
- Chapter 5 Linear Systems of Differential Equations 阅读
- Chapter 6 Numerical Methods 阅读
- Chapter 7 Nonlinear Systems and Phenomena 阅读
- Chapter 9. Eigenvalues and Boundary Value Problems 阅读
- Chapter 8. Fourier Series Methods 阅读
- 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
- Signal Processing & Electronics
- 6.01 Intro to EECS I
- 6.00 Introduction to Computer Science and Programming
- 6.100 A Introduction to Computer Science Programming in Python
- 6.0002 Introduction to Computational Thinking and data Science
- 6.02 Intro to EECS II
- UCB EE 16 A
- UCB EE 16 B
- 6.002 Circuits and Electronics
- 6.003 Signals and Systems
- UCB EE 120 Signals and Systems
- Data science
- Strang, Gilbert. Linear Algebra and Learning from Data. 阅读
- 18.065 Readings 阅读并总结，总结见此
- 18.065 Video Lectures 观看
- 18.065 Assignments 完成
- 18.065 Final Projects 完成
- mitmath/18065: 18.065/18.0651: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (github.com) 现代版 Notes 阅读
- mitmath/18065: 18.065/18.0651: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (github.com) Problem Sets 完成
- UCB Data 8
- UCB Data 100
- Harvard CS 109 A
- 18.05 Introduction to Probability and Statistics
- 6.3700 Introduction to Probability
- 6.3800 Introduction to Inference
- 18.05 Introduction to Probability and Statistics
- 18.600 Probability and Random Variables
- 18.650 Fundamentals of Statistics
- 6.042 J Mathematics for Computer Science
- Intro to Mathematical Reasoning
- Principles of discrete Applied mathematics
- 18.510 Introduction to Mathematical Logic and Set Theory
- 18.504 Seminar in Logic
- 18.515 Mathematical Logic
- 18.700 Advanced Linear Algebra
- Linear Algebra done right 阅读，阅读笔记见此
- Linear Algebra Done Right Videos (axler.net) Video Lectures 观看
- 18.700 Readings 完成并总结，总结见此
- 18.700 Assignments 完成并总结
- 18.700 Supplementary Notes 阅读并总结，总结见此
- 18.700 Course page (mit.edu) 现代版 Reading 完成并总结，总结见此
- 18.700 Course page (mit.edu) 现代版 Problem Sets 完成并总结，总结见此
- Math 115A (ucla.edu) 陶哲轩亲授，笔记见此 linear-algebra-notes.pdf (wordpress.com)
- 18.303 Linear Partial Differential Equations: Analysis and Numerics
- Strang, Gilbert. Computational Science and Engineering. 阅读并总结
- https://www.amazon.com/Introduction-Differential-Equations-Undergraduate-Mathematics/dp/3319020986 阅读并总结
- mitmath/18303: 18.303 - Linear PDEs course (github.com) Notes 阅读
- mitmath/18303: 18.303 - Linear PDEs course (github.com) Midterm 完成
- mitmath/18303: 18.303 - Linear PDEs course (github.com) Problem Sets 完成
- mitmath/18303: 18.303 - Linear PDEs course (github.com) Supplementary Notes 阅读并总结
- Syllabus | Linear Partial Differential Equations: Analysis and Numerics | Mathematics | MIT OpenCourseWare Lecture Summaries 阅读并总结
- Syllabus | Linear Partial Differential Equations: Analysis and Numerics | Mathematics | MIT OpenCourseWare Assignments 完成
- Syllabus | Linear Partial Differential Equations: Analysis and Numerics | Mathematics | MIT OpenCourseWare Exam 完成
- Syllabus | Linear Partial Differential Equations: Analysis and Numerics | Mathematics | MIT OpenCourseWare Projects 完成
- 高等代数
- 矩阵计算
- 矩阵分析
- 矩阵论
- 18.330 Introduction to Numerical Analysis
- 18.04 Complex Variables with applications
- 18.100A Real Analysis
- 18.100B Real Analysis
- 18.100P Real Analysis
- 18.100Q Real Analysis
- 18.S190 Introduction To Metric Spaces 或 18.S190, IAP 2023 (mit.edu)
- 18. S191 index — Interactive Computational Thinking — MIT
- 18.701 Algebra I
- 18.335[J] Introduction to Numerical Methods
- 6.7300[J] Introduction to Modeling and Simulation
- 16.920[J] Numerical Methods for Partial Differential Equations
- 18.085 Computational Science and Engineering I
- 18.336[J] Fast Methods for Partial Differential and Integral Equation
- 18. C 06 Linear Algebra and Optimization
- Linear Algebra and Optimization for Machine Learning 阅读
- cis515-13-sl1-a.pdf (upenn.edu)
- Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares (stanford.edu)
- Stanford 凸优化课程
- 18.204 Undergraduate Seminar in Discrete Mathematics
- 18.211 Combinatorial Analysis
- 6.120A Discrete Mathematics and Proof for Computer Science
- 6.1210 Introduction to Algorithms
- 6.045 J Automata, Computability, and Complexity
- 18.400[J] Computability and Complexity Theory
- 18.404 Theory of Computation
- 18.410[J] Design and Analysis of Algorithms
- 18.424 Seminar in Information Theory
- 18.434 Seminar in Theoretical Computer Science
- 18.453 Combinatorial Optimization
- 18.337[J] Parallel Computing and Scientific Machine Learning
- 18.338 Eigenvalues of Random Matrices
- 18.367 Waves and Imaging
- University Physics
- 《University Physics》阅读并做笔记，笔记见此
- Yale Fundamentals of physics I 学习
- Yale Fundamentals of physics II 学习
- Classical Mechanics 学习
- Physics II: Electricity and Magnetism 学习
- Relativity 学习
- Physics III: Vibrations and Waves 学习
- 18.415[J] Advanced Algorithms
- 18.416[J] Randomized Algorithms
- 18.425[J] Foundations of Cryptography
- 8.01 Physics I
- 8.02 Physics II
- 8.03 Physics III
- 8.04 Quantum Physics I
- 8.041 Quantum Physics I
- 8.05 Quantum Physics II
- 18.435[J] Quantum Computation
- 18.437[J] Distributed Algorithms
- 18.455 Advanced Combinatorial Optimization
- 18.702 Algebra II
- 18.703 Modern Algebra
- 18.783 Elliptic Curves
- 18.102 Introduction to Functional Analysis
- 18.112 Functions of a Complex Variable
- 18.303 Linear Partial Differential Equations
- 18.642 Topics in Mathematics with Applications in Finance
- 18.125 Measure Theory and Analysis
- 18.675 Theory of Probability
- 18.676 Stochastic Calculus
- 18.615 Introduction to Stochastic Processes
- 18.657 Topics in Statistics
- 18.900 Geometry and Topology in the Plane
- 18.212 Algebraic Combinatorics
- 18.901 Introduction to Topology
- 18.721 Introduction to Algebraic Geometry
- 18.781 Theory of Numbers
- 18.950 Differential Geometry
- 18.217 Combinatorial Theory
- 18.218 Topics in Combinatorics
- 18.225 Graph Theory and Additive Combinatorics
- 18.226 Probabilistic Methods in Combinatorics
- 18.705 Commutative Algebra
- 18.715 Introduction to Representation Theory
- 18.725 Algebraic Geometry I
- 18.745 Lie Groups and Lie Algebras I
- 18.704 Seminar in algebra
- 18.782 Introduction to Arithmetic Geometry
- 18.706 Noncommutative algebra
- 18.755 Lie Groups and Lie Algebras II
- 18.755 Lie Groups and Lie Algebras II
- 18.101 Analysis and Manifolds
- 18.102 Introduction to Functional Analysis
- 18.103 Fourier Analysis: Theory and Applications
- 18.104 Seminar in Analysis
- 18.112 Functions of a Complex variable
- 18.152 Introduction to Partial Differential Equations
- 18.950 Differential Geometry
- 18.994 Seminar in Geometry
- 18.125 Measure Theory and Analysis
- 18.155 Differential Analysis I
- 18.156 Differential Analysis II
- 18.952 Theory of Differential Forms
- 18.965 Geometry of Manifolds I
- 18.966 Geometry of Manifolds II
- 18.781 Theory of numbers
- 18.784 Seminar in Number Theory
- 18.785 Number Theory I
- 18.677 Topics in Stochastic Processes
- 18.655 Mathematical Statistics
- 18.904 Seminar in Topology
- 18.116 Riemann Surfaces
- 18.905 Algebraic Topology I
- 18.906 Algebraic Topology II
- 18.300 Principles of Continuum Applied mathematics
- 18.352[J] Nonlinear Dynamics: The Natural Environment
- 18.353[J] Nonlinear Dynamics: Chaos
- 18.354[J] Nonlinear Dynamics: Continuum Systems
- 18.384 Undergraduate Seminar in Physical mathematics
- 18.417 Introduction to Computational Molecular Biology
- 18.305 Advanced Analytic Methods in Science and Engineering
- 18.306 Advanced Partial Differential Equations with Applications
- 2.25 Fluid Mechanics
- 12.800 Fluid Dynamics of the Atmosphere and Ocean
- 18.355 Fluid Mechanics
- 18.357 Interfacial Phenomena
- 18.367 Waves and Imaging
- 8.07 Electromagnetism II
- 18.369[J] Mathematical Methods in Nanophotonics
- 2.003[J] Dynamics and Control I
- 18.075 Methods for Scientists and Engineers
- 18.376[J] Wave Propagation
- 18.377[J] Nonlinear Dynamics and Waves