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The book

The whole syllabus, straight through, 83 pages. Each one is a single idea: why the exam asks it, the explanation, a visual you can move, the trap it hides, and two questions to check you actually followed it. Free to read, no account needed.

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Unit 1 — Analysis & Linear Algebra

29 pages
  1. 1.Completeness, sup/inf, Archimedean property
  2. 2.Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy
  3. 3.limsup, liminf and subsequential limits
  4. 4.Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements
  5. 5.Continuity, uniform continuity, Lipschitz
  6. 6.Differentiability, mean value theorems, Taylor, L'Hôpital
  7. 7.Riemann integration and criteria
  8. 8.Improper integrals and convergence tests
  9. 9.Pointwise vs uniform convergence, M-test, Dini
  10. 10.Power series, radius of convergence, Abel's theorem
  11. 11.Arzelà–Ascoli and equicontinuity
  12. 12.Partial derivatives, differentiability, chain rule
  13. 13.Inverse and implicit function theorems, extrema
  14. 14.Open/closed sets, limit points, closure, interior
  15. 15.Compactness: open covers, sequential, Heine–Borel
  16. 16.Completeness and Baire category
  17. 17.Connectedness and path-connectedness
  18. 18.Measurable sets and functions
  19. 19.Lebesgue integral, MCT, DCT, Fatou
  20. 20.L^p spaces essentials
  21. 21.Bases, dimension, rank–nullity
  22. 22.Linear transformations, matrix representation, change of basis
  23. 23.Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton
  24. 24.Diagonalisability criteria
  25. 25.Jordan canonical form
  26. 26.Rational canonical form
  27. 27.Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem
  28. 28.Quadratic forms, positive definiteness, Sylvester's law
  29. 29.Determinants, trace, block matrices, rank inequalities

Unit 2 — Complex Analysis, Algebra & Topology

24 pages
  1. 30.Cauchy–Riemann equations, harmonic functions
  2. 31.Power series and analyticity
  3. 32.Cauchy's theorem and integral formula
  4. 33.Liouville, Morera, maximum modulus principle
  5. 34.Laurent series, classification of singularities, Casorati–Weierstrass
  6. 35.Residue theorem and standard contour integrals
  7. 36.Argument principle, Rouché's theorem, open mapping
  8. 37.Conformal maps, Möbius transformations, Schwarz lemma
  9. 38.Subgroups, cosets, Lagrange, cyclic groups
  10. 39.Normal subgroups, quotients, isomorphism theorems
  11. 40.Permutation groups: cycles, sign, conjugacy in S_n and A_n
  12. 41.Group actions, class equation, p-groups
  13. 42.Sylow theorems and groups of small order
  14. 43.Finite abelian groups
  15. 44.Ideals, quotient rings, prime & maximal ideals, CRT
  16. 45.Euclidean, PID, UFD hierarchy
  17. 46.Polynomial rings and irreducibility tests
  18. 47.Field extensions, splitting fields, finite fields
  19. 48.Galois theory essentials
  20. 49.Topological spaces, bases, subspace/product/quotient
  21. 50.Continuity, homeomorphism, separation axioms
  22. 51.Compactness and Tychonoff
  23. 52.Connectedness and components
  24. 53.Standard spaces: cofinite, cocountable, Sorgenfrey, Cantor set

Unit 3 — ODE, PDE & Applied Mathematics

16 pages
  1. 54.Existence–uniqueness, Picard, Lipschitz
  2. 55.Linear ODE, Wronskian, variation of parameters, systems
  3. 56.Sturm–Liouville problems and Green's functions
  4. 57.Stability and phase portraits
  5. 58.First-order PDE: Lagrange, Charpit, characteristics
  6. 59.Classification and canonical forms
  7. 60.Laplace, heat and wave equations: separation of variables
  8. 61.Root finding: bisection, Newton–Raphson, fixed point, order of convergence
  9. 62.Interpolation and numerical integration with error terms
  10. 63.Numerical ODE: Euler, Runge–Kutta
  11. 64.Euler–Lagrange equation and standard functionals
  12. 65.Isoperimetric problems
  13. 66.Fredholm and Volterra equations
  14. 67.Separable kernels and resolvent kernels
  15. 68.Lagrangian formalism and generalised coordinates
  16. 69.Hamiltonian formalism and conservation laws

Unit 4 — Probability & Statistics

14 pages
  1. 70.Axioms, conditional probability, independence, Bayes
  2. 71.Random variables, distributions, moments, MGF
  3. 72.Standard discrete and continuous distributions
  4. 73.Joint distributions, transformations, order statistics
  5. 74.Modes of convergence, WLLN, SLLN, CLT
  6. 75.Markov chains: classification of states, stationary distributions
  7. 76.Sufficiency, completeness, UMVUE, Cramér–Rao
  8. 77.MLE and method of moments
  9. 78.Neyman–Pearson lemma and UMP tests
  10. 79.Likelihood ratio and standard tests
  11. 80.Gauss–Markov, regression, ANOVA basics
  12. 81.Multivariate normal distribution
  13. 82.SRS, stratified and systematic sampling
  14. 83.CRD, RBD, LSD essentials
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