Syllabus

MA280 Science (honours)

MA286 Real Analysis

• Continuity and differentiability of a function f:R^m->R^n, partial derivatives, directional derivatives, the Chain Rule
• Maxima and minima
• Revision of the main definitions and properties of sequences and series of real numbers
• Lim inf and lim sup, Cauchy's criterion for convergence, Taylor series, power series, uniform convergence, differentiation term by term
• Multiple integrals

Texts

• T.M. Apostol, "Mathematical Analysis" (Addison Wesley)

MA287 Complex Analysis

• Functions of a complex variable: differentiability, the Cauchy-Riemann equations, harmonic conjugates
• Line integrals, logz and e^z
• Cauchy's Integral Theorem, Cauchy's Formula, Cauchy's Inequalities
• The Laurent series of a function, poles, residues
• Contour integration, Rouché's Theorem
• Conformal mappings, Möbius transformations

Texts

• R.V. Churchill & J.W. Brown, "Complex Variables and Applications" (McGraw Hill)
• M.R. Spiegel, "Complex Variables" (Schaum Outline)

MA283 Linear Algebra

• Vector spaces, bases, dimension, linear maps, matrix representation of linear maps
• Matrix algebra, kernels and images, least squares fitting, inner product spaces, the Gram-Schmidt process
• Fourier series, dual spaces, the rank of a matrix, determinants, eigenvalues and eigenvectors, the characteristic polynomial, quadratic forms
• Diagonalisation of a symmetric or Hermitian linear map
• Triangularisation of a linear map
• The Hamilton-Cayley theorem, linear programming

Texts

• T.S. Blyth & E.F. Robertson, "Matrices and Vector Spaces and Linear Algebra" (Chapman & Hall)
• S. Lipschutz, "Linear Algebra" (Schaum Outline)

References

• H. Anton, "Elementary Linear Algebra" (Wiley)
• J.W. Archbold, "Algebra" (Pitman)
• G. Birkhoff & S. MacLane, "A Survey of Modern Algebra" (Macmillan)
• I.N. Herstein, "Topics in Algebra" (Blaisdell)
• S. Lang, "Linear Algebra" (Springer)
• T.A. Whitelaw, "An Introduction to Linear Algebra" (Blackie)

MA284 Discrete Maths

• Enumeration: product rule, sum rule and sieve principle, selections and distributions, the pigeonhole principle
• Graphs, the fundamentals (including various notions of 'path' and 'tree'), plus a study of some of the following topics: colouring problems, bipartite graphs, Hamiltonian graphs, planar graphs and tournaments. Algorithms and applications are emphasised throughout.

Texts

• N.L. Biggs, "Discrete Mathematics", (Oxford)