A Mathematician's Survey

Mathematics
By OgbonLab

A graduate-level tour through the rooms most textbooks lock you out of.

A graduate-level overview: linear algebra, analysis, topology, differential equations, and more.

16 parts 77 sections Free, browser-native
Start reading → First up: Why Linearity Is Universal

Table of contents

Every section is a working session: text, math, code, interactive widgets. Click any title to jump in.

Part 1: Chapter 1: Linear Algebra Toolkit

  1. Why Linearity Is Universal
  2. Vectors in R^n and the Standard Basis
  3. Linear Maps and Matrix Representations
  4. Basis, Span, and Dimension
  5. The Determinant of a Square Matrix
  6. Rank-Nullity and Conditions for Invertibility
  7. Similar Matrices and Change of Basis
  8. Eigenvalues, Eigenvectors, and Diagonalization

Part 2: Chapter 2: Single-Variable Real Analysis

  1. Limits and Convergence
  2. Continuous Functions
  3. The Derivative
  4. The Riemann Integral
  5. The Fundamental Theorem of Calculus
  6. Sequences of Functions: Pointwise Convergence
  7. Uniform Convergence and Its Consequences

Part 3: Chapter 3: Calculus of Several Variables

  1. Vector-Valued and Multivariable Functions
  2. Limits and Continuity in R^n
  3. The Total Derivative as a Jacobian
  4. The Inverse Function Theorem
  5. The Implicit Function Theorem

Part 4: Chapter 4: Point-Set Topology Basics

  1. Open Sets, Closed Sets, and Topological Spaces
  2. The Standard Topology of R^n
  3. Metric Spaces and Distances
  4. Bases and Generation of Topologies

Part 5: Chapter 5: Integral Theorems of Vector Calculus

  1. Vector Calculus Preliminaries
  2. The Divergence Theorem
  3. Stokes' Theorem
  4. Physical Interpretations
  5. Proofs and Connections

Part 6: Chapter 6: Differential Forms and the Generalized Stokes Theorem

  1. Volumes of Parallelepipeds
  2. Differential Forms and Vector Fields
  3. The Exterior Derivative
  4. Manifolds
  5. The Generalized Stokes' Theorem

Part 7: Chapter 7: Differential Geometry, Curvature

  1. Curvature of Plane Curves
  2. Curvature and Torsion of Space Curves
  3. Gaussian and Mean Curvature of Surfaces
  4. The Gauss-Bonnet Theorem

Part 8: Chapter 8: Models of Geometry

  1. Euclidean Geometry and the Parallel Postulate
  2. Elliptic (Spherical) Geometry
  3. Hyperbolic Geometry
  4. Curvature and the Three Geometries

Part 9: Chapter 9: Complex Analysis, Holomorphic Functions

  1. Holomorphic Functions and Complex Differentiation
  2. The Cauchy-Riemann Equations
  3. Cauchy's Integral Formula
  4. Power Series and Analytic Functions
  5. Conformal Maps and Geometric Function Theory

Part 10: Chapter 10: Foundations, Cardinality, Choice, and Incompleteness

  1. Countability and Cardinal Numbers
  2. Russell's Paradox and the Foundations Crisis
  3. The Axiom of Choice and Its Equivalents
  4. Gödel's Incompleteness Theorems

Part 11: Chapter 11: Abstract Algebra Survey

  1. Groups: Axioms and First Examples
  2. Group Actions and Representations
  3. Rings, Ideals, and Quotient Rings
  4. Fields, Field Extensions, and Galois Theory

Part 12: Chapter 12: Measure Theory and the Lebesgue Integral

  1. Sigma-Algebras and Lebesgue Measure
  2. Sets of Measure Zero and the Cantor Set
  3. The Lebesgue Integral
  4. Monotone and Dominated Convergence

Part 13: Chapter 13: Fourier Series and Transforms

  1. Periodic Functions and Wave Phenomena
  2. Fourier Series Expansions
  3. Convergence Issues for Fourier Series
  4. The Fourier Transform on the Real Line

Part 14: Chapter 14: Differential Equations

  1. What Is a Differential Equation?
  2. Methods for Ordinary Differential Equations
  3. The Laplacian and Harmonic Functions
  4. The Heat Equation
  5. The Wave Equation

Part 15: Chapter 15: Combinatorics and Probability

  1. Counting Principles and Combinatorial Identities
  2. Probability: Sample Spaces and Events
  3. Conditional Probability and Independence
  4. Random Variables, Expectation, and Variance
  5. Limit Theorems and the Central Limit Theorem

Part 16: Chapter 16: Algorithms and Complexity

  1. Sorting Algorithms as a Concrete Starting Point
  2. Big-O and Asymptotic Complexity
  3. Graph Algorithms: BFS, DFS, Shortest Paths
  4. P, NP, and Computational Hardness

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