FACULTY OF ARTS HANDBOOK
It will be assumed that students attending this course have a knowledge of the work prescribed for Pure Mathematics at the Matriculation Examination.
( а) Preliminary reading: At least two of the following:
H.—Signpost to Mathematics. (ThriftBooks or Pitman.) Titchmarsh, E.
C. Mathematics for the General Reader. ( Hutchinson.)Dantzig, T.—Number,
The Language of Science. (Anchor.)
P.—Riddles in Mathematics. ( Pelican. )Sawyer, W.
W.Mathematician's Delight. ( Pelican. )(b) Prescribed textbooks:
R. First Course in Calculus. ( Wiley.)alternative.
B.—Calculus and Analytic Geometry.(Addison-Wesley.) } Weatherburn, C.
W. Elementary Vector Analysis. ( Bell.)} alternative.
S. Elementary Vector Geometry. ( Wiley. )
A book of mathematical tables. (Kaye and Laby,
Four-figure Mathematical Tables(Longmans.) will be provided in examinations. )
(c) Recommended for reference:
L.-Higher Algebra for Schools.(Oxford.)
McArthur, N., and Keith,
A, Intermediate Algebra. ( Methuen.)Cow, Margaret,
M. Pure Mathematics. (EnglishUniversities Press.) (d) Students who are aiming at honours may also use with profit:
L. Higher Algebra,the sequel, starting with ch. XV. (Oxford.) Durell, C. V.,
and Robson-Advanced Algebra,vols. 1 and 2. (Bell.)
Durell, C. V.,
Trigonometry. ( Bell. )EXAMINATION. Two 3-hour papers.
86. PURE MATHEMATICS PART II
A course of three lectures per week for half the year and two lectures per week in the remainder. together with practice classes throughout the year.
After the first term the course for day students may be divided into two alternative syllabuses, option A being devoted to the further study of calculus, option B to more fundamental studies. Only the former option will be given if the support offering for the latter, or the staff available to conduct it, is inadequate.
It is not necessary to signify which option will be desired until late in first term.
Functions.Exponential and related functions of a complex variable.
Differential Equations.Standard types of ordinary differential equations of the first and second orders. Linear differential equations, including solution by series and simultaneous systems.
Integrals.Infinite and improper integrals. Reduction formulae. Curvilinear integrals. Multiple integrals.
Functions of Seven Variables.Simple matrices. Analytical solid geometry.
Determinants. Directional derivatives. Stationary points. Envelopes. Change of variables. Polar coordinates. Surface integrals.
Series.Convergence. Absolute and conditional convergence. Power . series.
Taylor's theorem for functions of one variable. Approximate calculations with power series.
OptionВ: Topics will be chosen to replace the later work on differential equations, functions of two variables, and series. In former years topics have been selected from: euclidean geometry, non-euclidean geometry, elementary number theory, elementary theory of equations, theory of conics and orbits.
(a) Preliminary reading: At least two of the following.
W. Prelude to Mathematics.(Pelican.)
E.-What is Mathematics? ( O.U.P. ) Polya, G.—Mathematics and Plausible Reasoning.(Princeton.) Weyl, H.-Symmetry. (Princeton.)
(b) Prescribed textbooks: One of the following:
Calculus.(Wiley.) Maxwell, E.
A.—Analytical Calculus,vols. III, W. (C.U.P.)
B.—Calculus and Analytical Geometry.(Addison-Wesley.) Courant,
R. Differential and Integral Calculus,vols. I, II. (Blackie.), EXAMINATION. Two 3-hour papers.
87. PURE MATHEMATICS PART III-COURSE A
A course of tbree lectures per week, with practical classes, throughout the year.
Students who do sufficiently well in this course and in its examination may, if they make application be admitted to Pure Mathematics III honours course.
(i) Numerical Mathematics or an alternative assignment (to be done in the long vacation preceding the course; no lectures given) :
Numerical exercises on summation of series, difference tables, interpolation, integration, solution of differential equations, curve fitting, simultaneous linear equations and determinants.
Intending' students should obtain the exercises and instructions from the lathe matics department in January or February before the course begins and should hand in their work complete not later than 31 Mařch. Calculating machines will be available for this work, on request, and may be used in the Mathematics department.
(ii) Analysis ( two lectures per week, less one in third term) :
Sequences. Series of positive terms, absolute and conditional convergence.
Differentiable and continuous functions of one real variable. Convergence of infinite and improper integrals.
Double series, multiplication of series, partial fraction expansions.
Uniform convergence of series of functions. Power series, including the elementary functions of a complex variable. Fourier series.
Continuous functions of several variables. Functions defined by integrals. Multiple integrals.
( iii) Linear Algebra ( one lecture per week in the first two terms; or an equal number of lectures during one term) :'
Linear transformations. Matrix algebra. Characteristic polynomial. Quadratic forms. Systems of linear equations.
(iv) Special Functions (two lectures per week in third term):
Series solution of linear differential equations. Legendre polynomials. Bessel functions. Boundary value problems with linear partial differential equations.
(a) Recommended for preliminary reading:
Bell, E. T. Mathematics, Queen and Servant of Science. ( McGraw-Hill,) Kasner, E. and Newman, J. R.—Mathematics and the Imagination. ( Bell.) Courant, R, and Robbins, H. E.—What is Mathematics? ( O.U.P. )
Sawyer, W. W.—Prelude to Mathematics. ( Pelican.)
Sawyer, W. W, —A Concrete Approach to Abstract Algebra. (Freeman.) ( b ) . Prescribed textbooks:
Ferrar, W. L.—Textbook of Convergence. (О.U.Р.) Brand, L: Advanced Calculus. ( Wiley.)
Courant, R.—Differential and Integral Calculus. ( BIacide. ) } (altern.) Widder, D. V.—Advanced Calculus. ( Prentice-Hall.) JJJ . Aitken, A C.-Determinants and Matrices. ( Oliver &_ Boyd. )
Murdoch, D. C.—Linear Algebra for Undergraduates. (Wiley). . (altern.) Ayres, F.—Theorems and Problems of Matrices. (Schaum. )
Churchill, R. V.—Fourier Series and Boundary Value Problems. (McGraw-Hill. ) Sneddon, I. N.-Special Functions. (Oliver and Boyd.) . Rainville, E. D.—Special Functions. (Macmillan.)
Two 3-hour papers. Before admission to the examination candidates must have satisfactorily completed division (i) : Practical Mathematics.
88. PURE MATHEMATICS PART III—COURSE B
A course of three lectures per week, with practice classes, throughout the year.
This course is designed mainly for those who propose to take up school-teaching in mathematics subects; but it is also recommended for those who are interested in a logical and critical scrutiny of the foundations, and in mathematics as an element of general culture rather than in mathematics as a tool of trade. The intention of the course is to embed the subject-matter of school mathematics in a larger body of
FACULTY OF ARTS ØBOОK
knowledge, which in one direction covers foundations and systematic logical develop- ment, and in another direction gives some indication of the role of mathematics in science, culture and society.
A selection of topics from (i) to (viii), together with (ix) and (x):
(i) Elements of mathematical logic.
(ii) Algebra. Introduction to abstract algebra.
( ii ) Algebra. Theory of equations.
( iv) Geometry. Projective and non-euclidean geometry.
(v) Analysis. Convergence. Expansions in infinite series.
(vi) Calculus. Functions of a complex variable.
(vii) Statistics. Theory of probability. Statistical distributions. Elements of genetics.
( viii) Natural philosophy. Critical examination of the principles of mechanics.
(ix) Essays. Two essays will be prescribed in lectures.
( X ) Vacation reading. As prescribed below and in lectures.
(a) Preliminary reading: As for Pure Mathematics part III course A, and also Adler, I.—The New Mathematics. ( Mentor. )
Khin, F. Elementary Mathematics—Arithmetic, Algebra. Analysis. (Dover.) Мeserve, B. E.—Fundamental Concepts of Algebra. (Addison-Wesley.) Мeserve, B. E.—Fundamental Concepts of Geometry. (Addison-Wesley.) (b) Prescribed textbook:
Courant, R., and Robbins—What is MathematicsP (O.U.P.) EXAMINATION. Two 3-hour papers.
89. PURE MATHEMATICS PART III—COURSE C
A course of three lectures, with practical work of three hours, per week through- out the year. The course is designed for those wishing to specialize in computational theories and techniques.
Students may not receive credit, for degree purposes, for Pure Mathematics III and Applied Mathematics III as two different subjects if they take course C in Pure Mathematics III.
As large a selection as possible from the following topics.
(i) Matrix Algebra. Linear transformations. Matrices. Reciprocal matrices.
Numerical methods. Characteristic equation. Reduction of quadratic forms.
(ii) Solution of polynomial equations. Numerical methods.
( iii) Difference tables. Operator theory of differences. Differences of polynomials.
Polynomial interpolation. Numerical differentiation and integration. Inverse inter- polation. Sub-tabulation. Numerical summation of series.
(iv) Numerical solution of differential equations. Ordinary first order. Simul- taneous first order. Higher orders. One- and two-point boundary value problems.
Eigenvalue problems. Partial differential equations. Network methods.
(v) Second order linear differential equations. Legendre and Tchebychef polу- nomials. Bessel functions. Asymptotic expansions.
(vi) Data analysis. Smoothing. Fourier analysis. Orthogonal functions.
(vii) Automatic computing. Design of computers. Elementary programming.
(a) Preliminary reading; As for Pure Mathematics, part III course A.
(b) Prescribed textbooks:
Aitken, A. C.—Determinants and Matrices. (Oliver and Boyd.) Hartree, D. R.—Numerical Analysis. (O.U.P. )
Whittaker, E. T., and Robinson, G.—The Calculus of Observations. (Blackie.) National Physical Laboratory—Modern Computing Methods. (Notes on Applied
Science, No. 16.) (H.M. Stationery like.) (c) Recommended for reference:
Nautical Almanac Of&ce—Interpolation and Allied Tables. (1.1. Stationery like.)
University of Melbourne Computation Laboratory—Programming Manual.
Corne, L. J.—Shorter Six-Figure Tables. (Chambers: )
Fletcher, A., Miller and Rosenhead—Index to Mathematical Tables. (Scientific Computing Service. )
Two 3-hour papers. Before admission to the examination students must have satisfactorily completed the prescribed practical work.
90. GENERAL MATHEMATICS
Acourse of three lectures and one tutorial class per week throughout the year.
The course is designed for students of the less quantitative sciences, and others who may require more knowledge of elementary mathematical methods and their uses than they have acquired beforehand. It is not a suitable basis for mathematical studies beyond part I and will not be accepted as such without further work and the permis
sion of the head of the department of Mathematics.
Students who have not passed a Matriculation Mathematics subject in a recent year are strongly recommended to seek advice as to preparatory work from the Department in January.
The course will aim at covering a fairly wide range of topics selected from those set out below. While attention will be drawn to the meaning and import- ance of mathematical rigour, the degree to which finer points of argument will be pursued will be conditioned by the scope of the work to be covered. It is intended that the later parts of the syllabus will demonstrate as many applications as possible of the earlier parts, so that the student may see a number of elementary mathematical methods in action.
Algebra. Algebra as a means of generalizing and abstracting features of scientific problems. Complex numbers. Determinants. Finite differences and interpolation.
Geometry. Two-dimensional co-ordinate geometry; straight line and circle;
elementary properties of conics; tracing of miscellaneous curves. Three-dimensional co-ordinate geometry; straight line and plane; sphere and simple quadrics. Intro- duction to vectors.
Calculus. Elementary differentiation and integration with special reference to various curves; equations of tangents and normals; curvature. Partial differentiation.
Introduction to multiple integrals. Exponential, logarithmic and other simple series;
hyperbolic functions; Taylor series. Mean values. Approximations. Fourier series.
Differential equations. Ordinary differential equations of first order and degree;
second order linear equations with constant coefficients and other simple types.
Simplest partial differential equations.
Mechanics. Development and application of the principles
ofmechanics of a particle and of a system of particles.
Probability. Probability as degree of belief; probability and frequency. De- velopment and use of the basic probability theorems. Probability and scientific method.
(a) Preliminary reading: As for Pure Mathematics Part L (b) Prescribed textbooks:
Cow, Margaret I.—Pure Mathematics. (E.U.P.)
Kaye and Laby—Four-figure Mathematical Tables. (Longmans.) EXAMINATION. Two 3-hour papers.
94. APPLIED MATHEMATICS PART I
Acourse of two lectures and one tutorial class per week throughout the year- SYLLABUS
The principles of Dynamics, treated with the help of vector analysis, with ap- plication to the plane motion of particles (including motion under a central force) and rigid bodies.
FACULTY OF ARTS HANDBOOK
It will be assumed that students have a knowledge of the work prescribed for Pure Mathematics and Calculus and Applied Mathematics at the matriculation examination, and that they are concurrently studying Pure Mathematics part I or have previously passed that subject.
(a) Preliminary reading: At least two of the following:
Kline, M.—Mathematics in the Physical World. (Addison-Wesley.) Mach, E.—The Science of Mechanics. (Open Court) (Ch. 2 & 3.) Peierls, R. E.—The Laws of Nature. (Allen & Unwin.)
Abbott, A.—Flatland. (Dover.) Darwin, G. H.—The Tides. (Murray. ) Maxwell, J. C.—Matter and Motion. (Dover.)
• Perry, J.—Spinning Tops. (S.P.C.K. ) (b) Prescribed textbook:
• Bullen, K. E. Introduction to the Theory of Mechanics. (Science Press.) Synge, J. L., and Griffith, B. A.—Principles of Mechanics. (McGraw-Hill.) (c) Recommended for reference:
Weatherburn, C. E. Elementary Vector Analysis. (Bell.) EXAMINATION. Two 3-hour papers.
95. APPLIED MATHEMATICS PART II
A course of two lectures, with two hours practice class, per week throughout the year. It is hoped that evening lectures will be available every year from now on.
• It• will be assumed that students are concurrently studying Pure Mathematics part II or have previously passed that subject.
(i) Vector analysis and potential theory. Differential and integral calculus of scalar and vector functions of position, with applications to gravitational and electrostatic fields. . ..
(ii) Partial differential equations of mathematical physics. Laplace's equation in cartesian, cylindrical and spherical polar coordinates with typical applications.
Solution by separation of variables. Wave and heat conduction equations. Cartesian tensors. Principal axis transformation.
( iii) General dynamics. Elements of rigid dynamics in three dimensions.
Lagrange's equations. Small vibrations of discrete and continuous systems.
(a) For preliminary reading:
Bullen, K. E.—Theory of Mechanics, ch. XIII. ( Science Press.)
Weatherburn C. E. Elementarц Vector Analysis. (Bell), and the chapters on partial derivatives and the chain rule for functions of several variables in any standard Calculus book.
(b) Prescribed textbooks:
Hildebrand, F. B.—Advanced Calculus for Applications. (Prentice-Hall.) Synge, J. L., and Griffith, B. A.—Principies of Mechanics. (McGraw-Hill.) 1 alt.
Jaeger, J. C. Introduction to Applied Mathematics. ( O.U.P.) J
EXAMINATION. Two 3-hour papers. .
96. APPLIED MATHEMATICS PART III
Students may not receive credit, for degree purposes, for both this subject and Pure Mathematics part III, course C.
A course of three lectures per week throughout the year, together with practical work for three hours per week during about half the year.
(i) Numerical analysis and computation. Solution of linear and non-linear equations; inversion of matrices; finite differences with application to interpolation, differentiation, integration and summation of series; numerical
solution of ordinary and partial differential equations; curve-fitting and harmonic analysis.
(ii) Elementary hydrodynamics.
( iii) Cartеsiaп tensors and elementary elasticity.
(iv) Variational principles.
Students who studied Applied Mathematics Part II ( Pass) in years prior to 1962 will not have received item (i) of the present syllabus for that subject. As this knowledge will be assumed in lectures, they should carry out the following preliminary study:
Hildebrand, F. B.—Advanced . Calculus for Applications, Ch. 6. (Prentice- Hall ), possibly supplemented by
Weatherburn, C. E.—Advanced Vector Analysis, Ch. 1-2. ( London, Bell).
(a) Prescribed textbooks
Hartree, D. R: Numerical Analysis. ( O.U.P. ) (b) Recommended for reference:
Hildebrand, F. B.—Advanced Calculus for Applications. (Prentice-Hall.) Jeffreys, H., and B. S.—Methods of Mathematical Physics. (O.U.P.) Rutherford, D.. E.—Fluid Dynamics. ( Oliver and Boyd. )
Sokolnikoff, I. S. Mathematical Theory of Elasticity. ( McGraw-Iii.) Temple, G.-Cartesian Tensors. ( Methuen Monographs.)
Temple, G.—An Introduction to Fluid Dynamics. ( O.U.P. )
Maxwell, E. A.—Coordinate Geometry with Vectors and Tensors. (Oxford.) Weinstock, R. Calculus of Variations. ( McGraw-Hill. )
EXAMINATION. Two 3-hour papers.
HONOURS DEGREE D. SCHOOL OF MATHEMATICS ( For possible combinations with this school see p. 220.)
1. The course for B.A. with honours in Mathematics covers four years, during which the following subjects must be taken:
Pure Mathematics parts I, II, III, IV.
Applied Mathematics parts I, II, III, IV.
Also, candidates must take additional subjects (one of which must be Physics part I), so as to make up a total of eleven in all, and must qualify in Science French or Science German or Science Russian as prescribed for the B.Sc. degree, and must present a thesis on some approved topic in the final year. The full course will normally be as follows:
First Year: Pure Mathematics part I
Applied Mathematics part I . Physics part I
. *An Arts subject (see below). , Science Language
Second Year: Pure Mathematics part II Applied Mathematics part II Logic or Theory of Statistics part I Third Year: ' Pure Mathematics part III
Applied Mathematics part III Fourth Year: Thesis
Pure Mathematics part IV Applied Mathematics part IV
The details of the Mathematics subjects of this course are given below.
Students in combined honour courses which include Mathematics will take Pure Mathematics parts I, II, III, IV and the following provisions, so far as they are relevant, apply to them.
2. Students proposing to take the Second Year of the honour school of Mathe- matics should normally have obtained honours in Pure Mathematics part I and
*The fourth subject in First Year may be any Arts subject except thòse of Group 4 (c).
FACULTY OF ARTS HANDBOOK
Applied Mathematics part I. In exceptional circumstances students may qualify from a pass course; they will be advised what reading to undertake in the following long vacation so as to make up the additional ground that was covered in the honours lectures. It is most desirable that candidates should have a fair knowledge of Physics and some acquaintance with French and German.
Admission to the Second and higher years of the honour school must be approved by the faculty; candidates should make application as soon as possible after the examination results of the First Year have been published.
3. In the Fourth Year, candidates will carry out, under direction, a study of "a special topic, involving the reading and collation of the relevant mathematical litera- ture, and will present a thesis embodying this work. The topic will be chosen, in consultation with the staff of the department, at the beginning of the first term, and the thesis will be presented at the beginning of the third term. The thesis will be taken into account in determining the class list for the final examination.
4. The examinations in Pure Mathematics part III and Applied Mathematics part Ill (two papers in each), held at the end of the Third Year, will count as the first section of the final examination. The second section of this examination, held at the end of the Fourth Year, will cover the work of that year ( two papers in each of Pure and Applied Mathematics part IV), and will include also two general papers.
The results in both sections will be taken into account in determining the class list.
5. At the final examination the Wyselaskie Scholarship of 2,173 in Mathematics is awarded. This award may be held in conjunction with a University research grant.
Normally the Wyselaskie scholar will be required to pursue study or research in Mathematics or some other subject. See Calendar, regulation 6.7.
6. For students majoring in Mathematics who wish to pursue Physics or Chemistry to part II level the B.Sc. degree is available under the provisions of section 9, regulation 3.20, in the Calendar. Such students may further roceed to the degree of B.Sc. with honours on completing the Fourth Year of the honours school of Mathematics.'
7. The Professor Wilson Prize and the Professor Nanson Prize are awarded in alternate years for the best original memoir in Pure or Applied Mathematics. Candi- dates must be graduates of not more than seven years' standing from Matriculation.
See regulation 6.72(2) and (14) in the University Calendar.
Students are expected to read ( especially during the summer vacations) substantial portions of at least two of the books listed under "Preliminary Reading
for the several subjects. Many of the books are available in paperback editions.
In addition, attention is called to the following books on the history of mathematics.
Struik, D. J.—Concise History of Mathematics. (Dover.) Turnbuil, H. W.—The Great Mathematicians. (Methuen.) Bell, E. Т.—Men of Mathematics. (Pelican.)
Sarton, G. History of Mathematics. (Dover.) looper, A. Makers of Mathematics. (Faber.) van der Waerden—Science Awakening. (Groningen.) Dantzig, T. Bequest of the Greeks. (Allen & Unwin.) Boyer, C. B. Нistoщi/ of the Calculus. (Dover. )
85. PURE MATHEMATICS PART I
86. PURE MATHEMATICS PART II (Ions)
A course of four Iectures per week in the firšt two terms, and three in the third term, with tutorial work.
Ø course may be taken by those who have obtained adequate honours in Pure Mathematics I and by those who have passed satisfactorily In Pure Маthе mаtics II.
(i) Numerical Mathematics
oran alternative assignment (to
bedone in the long vacation preceding the course; no lectures given);