Syllabus  MA180
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 x^{n}", 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 A^{n}, systems
of recurrence relations, linear transformations

The integers: axioms for Z, wellordering, 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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