Volume 2: The maths-major level

Read the units in order. Each one says what it builds on. Inside a unit, each chapter opens with a short check of what it needs.

1. 0

   Thinking in proofs

   How to read a theorem, prove it, and check your own proof. Start here.

   • 0.1 How to read a theorem
   • 0.2 Proof techniques
   • 0.3 Sets, functions, and counting infinity
   • 0.4 Equivalence relations and partitions
   • 0.5 Integers mod n, gcd and Bézout
   • 0.6 Complex numbers and polynomials, done properly
   • 0.7 Writing a proof a grader accepts
2. 1

   What calculus really means

   The real numbers, limits, the derivative and the integral, made exact. Builds on Unit 0.

   • 1.1 The real numbers and completeness
   • 1.2 Sequences and limits, from zero
   • 1.3 Limits and continuity with epsilon and delta
   • 1.4 The derivative as best linear approximation
   • 1.5 The Riemann integral: area as a limit
   • 1.6 Series, power series, and uniform convergence
3. 2

   Linear algebra

   Vectors, linear maps and matrices: the language of almost everything. Builds on Unit 0.

   • 2.1 Vector spaces and subspaces
   • 2.2 Span, independence, basis, dimension
   • 2.3 Linear maps, null space, range, rank–nullity
   • 2.4 Inverses, one-to-one maps, isomorphism
   • 2.5 Matrices as maps, multiplication, change of basis
   • 2.6 Systems of equations and elimination
   • 2.7 Determinants
   • 2.8 Eigenvalues and eigenvectors
   • 2.9 Inner products, orthogonality, Gram–Schmidt, projection
   • 2.10 Least squares and regression as projection
   • 2.11 Markov chains and PageRank
4. 3

   Linear algebra, deeper

   The spectral theorem, the SVD, Jordan form, and the bridge to algebra. Builds on Unit 2.

   • 3.1 Symmetric matrices and the spectral theorem
   • 3.2 Quadratic forms and principal axes
   • 3.3 Orthogonal matrices, rotations and reflections
   • 3.4 The singular value decomposition and PCA
   • 3.5 Matrix norms, conditioning, and why numerics fail
   • 3.6 Cayley–Hamilton, Schur, and the Jordan form
   • 3.7 Dual spaces and quotient spaces
   • 3.8 Matrix calculus and backpropagation
   • 3.9 Groups, rings and fields: a preview built from what you know
5. 4

   Space and closeness

   Distance, completeness, compactness: the bridge to analysis and topology. Builds on Unit 1, Unit 2.

   • 4.1 Metric and normed spaces; open and closed sets
   • 4.2 Convergence, completeness and the contraction mapping theorem
   • 4.3 Compactness
   • 4.4 Connectedness and path-connectedness
   • 4.5 Continuity between metric spaces
6. 5

   Differential calculus in many variables

   The derivative as the best linear approximation, in any dimension. Builds on Unit 1, Unit 2, Unit 4.

   • 5.1 Scalar and vector fields, limits and continuity
   • 5.2 Directional derivatives, partial derivatives and the gradient
   • 5.3 The total derivative, the Jacobian matrix and tangent planes
   • 5.4 The chain rule for fields
   • 5.5 Higher partial derivatives and the equality of mixed partials
   • 5.6 Taylor's formula, the Hessian and extrema
   • 5.7 Inverse and implicit functions
   • 5.8 Constrained extrema and Lagrange multipliers
   • 5.9 Partial differential equations
   • 5.10 Gradient descent, momentum and Newton's method
7. 6

   Integration in many variables

   Line, surface and volume integrals, and the theorems that connect them. Builds on Unit 5.

   • 6.1 Paths and line integrals
   • 6.2 Gradient fields and potentials
   • 6.3 Multiple integrals
   • 6.4 Green's theorem and change of variables
   • 6.5 Surfaces, area and flux
   • 6.6 Curl, divergence, Stokes' theorem and Gauss's theorem
   • 6.7 Differential forms and the one big theorem
   • 6.8 Maxwell's equations, conservation laws and heat
8. 7

   Differential equations

   Change described by equations, solved with linear algebra. Builds on Unit 3, Unit 4, Unit 5.

   • 7.1 Existence and uniqueness for differential equations
   • 7.2 Linear differential equations
   • 7.3 The Wronskian, variation of parameters and undetermined coefficients
   • 7.4 Systems of equations and the matrix exponential
   • 7.5 Power series solutions, Legendre and Bessel
   • 7.6 Phase portraits, stability and linearization
   • 7.7 Applications to springs, circuits, epidemics and qubits
9. 8

   Fourier analysis

   Every function as a sum of waves, and the boundary-value problems they solve. Builds on Unit 1, Unit 2, Unit 5.

   Chapters coming.

10. 9

    Complex analysis

    Calculus with complex numbers, where one derivative forces infinitely many. Builds on Unit 1, Unit 5, Unit 6.

    Chapters coming.

11. 10

    Probability and statistics

    Randomness made exact, and how to learn from data. Builds on Unit 1, Unit 2.

    • 10.1 Probability as measure
12. 11

    Discrete mathematics

    Counting, recurrences and graphs. Builds on Unit 0.

    Chapters coming.

13. 12

    Number theory and algebra

    Primes and congruences, then groups, rings and fields. Builds on Unit 0, Unit 3.

    Chapters coming.

14. 13

    Geometry and topology

    Euclid to curved space, and the shape of spaces. Builds on Unit 4, Unit 5, Unit 6.

    Chapters coming.

15. 14

    Logic, sets and computation

    What a proof is, how big infinity gets, and what cannot be computed. Builds on Unit 0.

    Chapters coming.

16. 15

    Optimisation, dynamics and computation

    Finding the best choice, what happens when a rule is repeated forever, and how computers do mathematics. Builds on Unit 1, Unit 2, Unit 5, Unit 7.

    • 15.1 Numerical analysis