Advanced · Mathematics (Analysis & Algebra)

Advanced Mathematics Syllabus

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Block A

Differential and Integral Calculus

1. The Rigorous (Epsilon-Delta) Definition of a Limit

2. Limit Theorems and the Continuity of Functions

3. The Derivative as a Limit: Differentiability versus Continuity

4. Mean Value Theorems (Lagrange and Cauchy)

5. L'Hopital's Rule and Improper Limits

6. Optimisation and Curve Sketching with Derivatives

7. The Riemann Integral: Darboux Sums and Integrability

8. The Fundamental Theorem of Calculus

9. Advanced Integration Techniques (Partial Fractions, Euler Substitutions)

10. Improper Integrals of the First and Second Kind

11. Geometric Applications of Definite Integrals

12. Physical Applications of Integrals (Centre of Mass, Moment of Inertia)

13. Introduction to Functions of Several Variables (Domain, Limits)

14. Partial Derivatives and the Tangent Plane

15. Extrema of Multivariable Functions and Lagrange Multipliers

Block B

Real Analysis and Topology

1. Axioms of the Real Numbers, Supremum and Infimum

2. Metric Spaces and the Topology of the Real Line

3. Open, Closed and Compact Sets (the Heine-Borel Theorem)

4. Complete Spaces and the Banach Fixed-Point Theorem

5. Sequences and Series of Functions

6. Pointwise and Uniform Convergence

7. Differentiation and Integration of Series of Functions

8. Power Series, the Radius of Convergence and Taylor Series

9. Analysis on Manifolds: an Introduction to Differential Forms

10. Multiple Integrals and Fubini's Theorem

11. Line Integrals of Scalar and Vector Fields

12. Surface Integrals and the Flux of a Vector Field

13. Green's Theorem in the Plane

14. The Stokes and Gauss-Ostrogradsky Theorems

15. Introduction to Measure Theory (the Lebesgue Measure)

Block C

Linear and Abstract Algebra

1. Number Fields and the Rigorous Definition of a Vector Space

2. Subspaces, Linear Combinations and the Linear Span

3. Basis, Dimension and Coordinates in a Basis

4. Linear Maps, Kernel and Image (the Rank-Nullity Theorem)

5. The Matrix of a Linear Map and Change of Basis

6. Determinant Theory (Permutations, Laplace Expansion)

7. Eigenvalues, Eigenvectors and the Characteristic Polynomial

8. Diagonalisation of Matrices and Symmetric Matrices

9. Inner Product Spaces and Gram-Schmidt Orthogonalisation

10. Orthogonal and Unitary Matrices, the Singular Value Decomposition

11. Group Theory: Isometry Groups and Normal Subgroups

12. Group Homomorphisms and the Isomorphism Theorems

13. Rings and Ideals (Polynomial Rings)

14. Fields and Field Extensions

15. Introduction to Galois Theory

Block D

Differential Equations and Complex Analysis

1. Modelling with First-Order Differential Equations

2. Separable, Homogeneous and Linear First-Order Equations

3. Exact Equations and the Integrating Factor

4. Higher-Order Linear Equations with Constant Coefficients

5. Undetermined Coefficients and Variation of Parameters

6. The Laplace Transform and Its Applications to ODEs

7. Systems of Linear Differential Equations and the Phase Plane

8. Introduction to PDEs (the Wave and Heat Equations)

9. Analytic Functions and the Cauchy-Riemann Equations

10. Exponential, Trigonometric and Logarithmic Functions in the Complex Plane

11. Complex Integration and Cauchy's Integral Theorem

12. Cauchy's Integral Formula and Its Consequences (Liouville's Theorem)

13. Laurent Series and the Classification of Singularities

14. The Residue Theorem and the Evaluation of Real Integrals

15. Conformal Mappings