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MathematicsYear 3Quality review 0/4

Mathematics — Year 3

A rigorous third-year pure-mathematics course spanning real analysis, abstract algebra, an introduction to complex analysis, and first steps in partial differential equations. Every idea arrives in two modes — precise Theory and intuition-first Visual — with worked proofs, grounded analogies and plenty of practice.

≈ 13 h 25 min8 sections · 32 lessonsNo ratings yet
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What you'll learn

  • Apply the ε–N and ε–δ definitions to establish limits of sequences, continuity of functions and the convergence of series, and reason about the completeness of the real numbers via suprema and infima
  • Verify the group, ring and field axioms and work confidently with subgroups, cyclic groups, homomorphisms, Lagrange's theorem and modular arithmetic
  • Manipulate complex numbers geometrically, test analyticity with the Cauchy–Riemann equations, and use Euler's formula and polar form
  • Classify partial differential equations by order and linearity, and solve first-order transport equations and the heat equation by separation of variables

Before you start

  • Single- and multivariable calculus: limits, derivatives and integrals handled fluently
  • Linear algebra: vectors, matrices and systems of linear equations
  • Comfort reading and writing basic mathematical proofs (direct, contradiction, induction)

Academic references

Use the latest available edition and your institution's prescribed text.

  • OpenStax, Calculus, Volumes 1–3
  • Gilbert Strang, Introduction to Linear Algebra
  • Daniel J. Velleman, How to Prove It

Formal work standard

  • State definitions and hypotheses before using a theorem.
  • Separate proof, counterexample, and numerical evidence.
  • Complete the problem sheet without solution steps before reviewing errors.

Course content

0/32 lessons
1. Real Analysis4 lessons
  1. Sequences and the ε–N LimitLesson25 min
  2. Completeness, Suprema and InfimaLesson25 min
  3. Continuity and the ε–δ DefinitionLesson30 min
  4. Infinite Series and ConvergenceLesson25 min
2. Abstract Algebra: Groups, Rings and Fields4 lessons
  1. Groups and Their AxiomsLesson25 min
  2. Subgroups, Cyclic Groups and LagrangeLesson25 min
  3. Homomorphisms and IsomorphismsLesson25 min
  4. Rings and FieldsLesson25 min
3. Complex Analysis (Introduction)3 lessons
  1. Complex Numbers and the Argand PlaneLesson20 min
  2. Analytic Functions and Cauchy–RiemannLesson30 min
  3. The Complex Exponential and Euler's FormulaLesson25 min
4. Partial Differential Equations (Introduction)3 lessons
  1. What Is a PDE? Order and LinearityLesson20 min
  2. First-Order PDEs: The Transport EquationLesson25 min
  3. Separation of Variables and the Heat EquationLesson30 min
5. Real Analysis: Metric and Functional Foundations5 lessons
  1. Metric Spaces and Cauchy SequencesLesson25 min
  2. Compactness and Sequential CompactnessLesson25 min
  3. Uniform ContinuityLesson25 min
  4. Uniform Convergence of Function SequencesLesson25 min
  5. Riemann Integration and Fundamental ResultsLesson25 min
6. Abstract Algebra: Quotients, Actions and Fields5 lessons
  1. Normal Subgroups and Quotient GroupsLesson25 min
  2. Group Actions and Orbit–StabiliserLesson25 min
  3. Ideals and Quotient RingsLesson25 min
  4. Polynomial Rings and IrreducibilityLesson25 min
  5. Field Extensions and Algebraic ElementsLesson25 min
7. Complex Analysis: Integration and Residues4 lessons
  1. Contour IntegralsLesson25 min
  2. Cauchy's Theorem and Integral FormulaLesson25 min
  3. Taylor and Laurent SeriesLesson25 min
  4. The Residue TheoremLesson25 min
8. Partial Differential Equations: Classification and Boundary Problems4 lessons
  1. Classifying Second-Order PDEsLesson25 min
  2. The Wave EquationLesson25 min
  3. Laplace's Equation and Harmonic FunctionsLesson25 min
  4. Fourier Series and Boundary-Value ProblemsLesson25 min