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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)

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