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MA181 Analysis (part one)
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Review of differential and integral calculus
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Construction of R, countable and uncountable sets
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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"
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Tests for convergence of series, power series, products of series
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Use of symbolic computational software (Maple, Mathematica, Cayley, GAP)
MA182 Analysis (part two)
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Continuous and discontinuous functions, the Intermediate Value Theorem,
inverse functions
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Differentiability, the Chain Rule, the Mean Value Theorem, Taylor's Theorem
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Riemann integration
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The Fundamental Theorem of Calculus
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Series, the exponential and logarithmic functions
Texts
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F. Ayres & E. Mendelson. Calculus. Schaum Outline.
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M. Spivak, "Calculus" (Addison Wesley)
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J.C. Burkill, "A First Course in Mathematical Analysis" (Cambridge)
MA183 Algebra (part one)
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Basic matrix algebra for 2 by 2 and 3 by 3 matrices: multiplication, adjoint,
determinant, inverse, eigenvalues and eigenvectors
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Algebra for 2 by 2 matrices only: calculation of An, systems
of recurrence relations, linear transformations
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The integers: axioms for Z, well-ordering, induction; the division
algorithm, greatest common divisor, linear Diophantine equations; primes
and factorisation
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Functions and sets: composition; injective, surjective, bijective and inverse
functions; finite and countably infinite sets
MA184 Algebra (part two)
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Arithmetic modulo m, solution of congruences, applications
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Introduction to groups, rings and fields: examples from earlier topics,
cyclic groups, Lagrange's Theorem, polynomials
Texts
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N.L. Biggs. "Discrete Mathematics" (Oxford)
References
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R.B.J.T. Allenby, "Rings, Fields and Groups" (Arnold)
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J.W. Archbold, "Algebra" (Pitman)
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G. Birkhoff & S. MacLane, "A Survey of Modern Algebra" (Macmillan)
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J. Hunter et al, "Algebra and Number Theory" (Blackie)
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I.N. Herstein, "Topics in Algebra" (Blaisdell)
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W. Ledermann, "Introduction to the Theory of Finite Groups" (Oliver &
Boyd)
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T.A. Whitelaw, "An Introduction to Abstract Algebra" (Blackie)
Back to 1st Year Syllabus.
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