Faculty of Mathematic Handbook,1987 - University of Newcastle

662304 Topic L .Analysis of Metric Spaces· M.J. Hayes
Corequisite CO
Hours 1 lecture hour per week and 1 tutorial hour per
fortnight
Examination
Content
One 2-hour paper
Examples of metric spaces and normed vector spaces. Convergence of sequences,
continuity of maps. Limit points, closed and open sets. Compactness and application to
existence of maxima, uniform continuity and integrability of continuous functions, and
continuity of inverse functions. Completeness, contraction maps, Picard's theorem for
differential equations. Uniform convergence, differentiation and integration of sequences
and series, power series. Abel's limit theorem, Taylor series, Weierstrass approximation
theorem. Fourier series, convergence theorems, Gibb's phenomenon.
Text
References
Bartle, RG.
Giles, lR.
Goldberg, RR.
Simmons, G.F.
White, A.J.
Nil
PART III MATHEMATICS TOPICS
The Elements of Real Analysis (Wiley 1976)
Analysis of Metric Spaces (Uni. of Newcastle 1975)
Methods of Real Analysis (Ginn BlaisdeI11964)
Introduction to Topology and Modern Analysis
(McGraw-Hill 1963)
Real Analysis (Addison-Wesley 1968)
663101 Topic M - General Tensors and Relativity - P.K. Smrz
Prerequisite Topic CO
Hours
Examination
Content
2 lecture hours and 1 tutorial hour
per week for 1st half year
One 2-hour paper
Covariant and contravariant vectors, general systems of coordinates. Covariant
differentiation, differential operators in general coordinates. Riemannian geometry,
metric, curvature, geodesics. Applications of the tensor calculus to the theory of
elasticity, dynamics, electromagnetic field theory, and Einstein's theory of gravitation.
Text
References
Abram, J.
Landau, L.D. & Lifshitz, E.M.
Lichnerowicz, A.
Tyldesley, J.R.
Willmore, TJ.
Nil
Tensor Calculus through Differential Geometry
(Butterworths 1965)
The Classical Theory of Fields (pergamon 1962)
Elements of Tensor Calculus (Methuen 1962)
An Introduction to Tensor Analysis (Longman 1975)
An Introduction to Differential Geometry (Oxford 1972)
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663102 Topic N • Variational Methods and Integral Equations - C.l Ashman
Prerequisite Topic CO
Hours
I lecture hour per week
and 1 tutorial hour per fortnight
Examination
One 2-hour paper
Content
Problems with fixed boundaries: Euler's equation, other governing equations and their
solutions: parametric representation. Problems with movable boundaries: transversality
condition; natural boundary conditions; discontinuous solutions; comer conditions.
Problems with constraints. Isoperimetric problems. Direct methods. Fredholm's
equation; Volterra's equation; existence and uniqueness theorems; method of successive
approximations; other methods of solution. Fredholm's equation with degenerate kernels
and its solutions.
Text
References
Arthurs, A.M.
Chambers, L.G.
Elsgolc, L.E.
Kanwal, RP.
Weinstock, R.
Nil
Complementary Variational Principles (pergamon 1964)
Integral Equations: A Short Course (International 1976)
Calculus of Variations (Pergamon 1963)
Linear Integral Equations (Academic 1971)
Calculus of Variations (McGraw-Hill 1952)
663103 Topic 0 - Mathematical Logic and Set Theory - M.J. Hayes
Prerequisite
Hours
Examination
Content
Topics K & L are recommended but not essential,
but some maturity in tackling axiomatic systems
is required.
1 lecture hour per week and one tutorial per
fortnight
One 2-hour paper
The problem of the "number" of elements in an infinite set; paradoxes. Tautologies and
an axiomatic treatment of the statement calculus. Logically valid formula and an
axiomatic treatment of the predicate calculus. Fi .... st order theories, consistency,
completeness. Number theory. Goedel's incompleteness theorem. Set theory, axiom of
choice, Zorn's lemma.
Text
References
Crossley, J. et al.
Halmos, P.R.
Hofstadter, D.R.
Kline, M.
Lipschutz, S.
Margaris, A.
Mendelson, E.
Nil
What is Mathematical Logic? (Oxford 1972)
Naive Set Theory (Springer 1974; Van Nostrand 1960)
Godel, Escher, Bach: an Eternal Golden Braid
(penguin 1981)
The Loss of Certainty (Oxford 1980)
Set Theory and Related Topics (Schaum 1964)
First Order Mathematical Logic (Blaisdell 1967)
Introduction to Mathematical Logic 2nd edn
(Van Nostrand 1979, paperback)
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