Syllabus

MA180 Science (honours)

MA181 Analysis (part one)

• Review of differential and integral calculus
• Construction of R, countable and uncountable sets
• Sequences and limits, limits of sums and products, the geometric series "Sum from 1 to infinity of xⁿ", and the Dirichlet series "Sum from 1 to infinity of (1/x)ⁿ"
• Tests for convergence of series, power series, products of series
• Use of symbolic computational software (Maple, Mathematica, Cayley, GAP)

MA182 Analysis (part two)

• Continuous and discontinuous functions, the Intermediate Value Theorem, inverse functions
• Differentiability, the Chain Rule, the Mean Value Theorem, Taylor's Theorem
• Riemann integration
• The Fundamental Theorem of Calculus
• Series, the exponential and logarithmic functions

Texts

• F. Ayres & E. Mendelson. Calculus. Schaum Outline.
• M. Spivak, "Calculus" (Addison Wesley)
• J.C. Burkill, "A First Course in Mathematical Analysis" (Cambridge)

MA183 Algebra (part one)

• Basic matrix algebra for 2 by 2 and 3 by 3 matrices: multiplication, adjoint, determinant, inverse, eigenvalues and eigenvectors
• Algebra for 2 by 2 matrices only: calculation of Aⁿ, systems of recurrence relations, linear transformations
• The integers: axioms for Z, well-ordering, induction; the division algorithm, greatest common divisor, linear Diophantine equations; primes and factorisation
• Functions and sets: composition; injective, surjective, bijective and inverse functions; finite and countably infinite sets

MA184 Algebra (part two)

• Arithmetic modulo m, solution of congruences, applications
• Introduction to groups, rings and fields: examples from earlier topics, cyclic groups, Lagrange's Theorem, polynomials

Texts

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

References

• R.B.J.T. Allenby, "Rings, Fields and Groups" (Arnold)
• J.W. Archbold, "Algebra" (Pitman)
• G. Birkhoff & S. MacLane, "A Survey of Modern Algebra" (Macmillan)
• J. Hunter et al, "Algebra and Number Theory" (Blackie)
• I.N. Herstein, "Topics in Algebra" (Blaisdell)
• W. Ledermann, "Introduction to the Theory of Finite Groups" (Oliver & Boyd)
• T.A. Whitelaw, "An Introduction to Abstract Algebra" (Blackie)