MATHEMATICS
(ii) Calculus. Differentiation and integration, with the usual applications. The standard elementary functions. Introductions to infinite series and differential equations.
( iii) Additional topics may be given, selected from: complex numbers, conic sections, functions of two variables, theory of sets.
It will be assumed that students attending this course have passed the subject Pure Mathematics at the Matriculation Examination.
BOOKS
(a) Preliminary reading: At least two of the following:
Read, A. Н.—Signpost to Mathematics. ( Thrift Books or Pitman.) Titchmarsh, E. C.—Mathematics for the General Reader. ( Hutchinson.) Dantzig, T.-Number, The Language of Science. (Anchor.)
Northrop, E. P.—Riddles in Mathematics. ( Pelican.) Sawyer, W. W.—Mathematician's Delight. (Pelican.) ( b ) Prescribed textbooks:
Cooley, H. R. First Course in Calculus. ( Wiley.) alternative.
Thomas, G. B.—Calculus and Analytic Geometry. (Addison-Wesley.) $ Weatherburn, C. E. Elementary Vector Analysis. ( Bell.) alternative.
Schuster, 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:
Ferrar, W. L.-Н gheт Algebra for Schools. (Oxford.)
McArthur, N., and Keith, A. Intermediate Algebra. (Methuen.) Cow, Margaret, M. Pure Mathematics. (English Universities Press. ) (d) Students who are aiming at honours may also use with profit:
Ferrar, W. 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., and Robson—Advanced Trigonometry. (Bell.) EXAMINATION. Two 3-hour papers.
86.
PURE MATHEMATICS PART II
A course of three lectures per week 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 В to more fundamental studies. Іnју 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.
SYLLABUS
Complex 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.
Integrals. Infinite and improper integrals. Reduction formulae. Curvilinear integrals. Multiple integrals.
Functions of Several Variables. Simple matrices. Analytical solid geometry.
Determinants. Directional derivatives. Stationary points. Lagrange multiplier method.
Change of variables. Polar co-ordinates. Surface integrals.
Series. Convergence. Absolute and conditional convergence. Power series.
Taylor's theorem for functions of one variable. Approximate calculations with power series.
Introductions to set theory and group theory.
Option В: 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.
Books
(a) Preliminary reading: at Ieast two of the following:
Sawyer, W. W. Prelude to Mathematics. (Pelican.)
Courant, R., and Robbins, H. E.—What is Mathematics? (O.U.P. ) 139
FACULTY OF ARTS HANDBOOK Роhа, G. How to Solve it. (Anchor.)
Pedoe, D.—The Gentle Art of Mathematics. (Pelican.)
Reid, C. Introduction to Higher Mathematics. (Routledge & Kegan Paul.) (b) Prescribed textbooks:' One of the following:
Cooley, H. R. First Course in Calculus. (Wiley.)
Maxwell, E. A.—Analytical Calculus, vols. III and IV. ( C.U.P. )
Thomas, G. B.-Calculus and Analytical Geometry. (Addison-Wesley.) Courant, R.—Differential and Integral Calculus, vols. I and II (Blackie.) EXAMINATION. Two 3-hour papers.
87. PURE MATHEMATICS PART III—COURSE A
A course of three 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.
SYLLABUS
(i) Numerical Mntbеmaućs or Theory of Numbers or an alternative assignment, to be done in the long vacation preceding the course; no lectures given. Intending students should obtain the exercises and instructions from the Mathematics depart-.
ment in December or January before the course begins and should hand in their work complete not later than 31 March. This work will carry some weight in the examina-
tion. - - -
Numerical Mathematics. Exercises on summation of series difference tables, interpolation, integration, solution of differential equations, curve fitting, simultaneous linear equations and determinants. Calculating machines will be available for this work, on request, and may be used in the Mathematics department. It will be neces- sary to obtain, and to read relevant parts of:
Noble, B. Numerical Methods, vols. I and II. ( Oliver & Boyd.)
Theory of Numbers. Exercises on prime numbers, factorization, congruences,.
quadratic residues, continued fractions, Diophantine approximation, quadratic forms.
In addition to the exercises and instructions referred to above, it will be necessary to obtain and read:
Davenport, 1.—The Higher Arithmetic. ( Hutchinson.) (ii) Analysis (about 40 lectures):
Convergence of sequences. Series; absolute and conditional convergence. Multi- plication of series.
Continuous and diffетепtiаЫе functions of one real variable. Convergence of infinite and improper integrals.
Continuous functions of several variables; implicit functions. Functions defined by integrals. Multiple integrals.
Uniform convergence of series of functions. Power series, including the elementary functions of a complex variable. Fourier series.
( iii) Linear Algebra ( about 20 lectures )
Linear transformations. Matrix algebra. Characteristic polynomial; eigenvalues.
Quadratic forms. Systems of linear equations.
(iv) Special Functions (about 10 lectures):
Boundary value problems with linear partial differential equations. Series solution of linear differential equations. Legendre polynomials. Bessel functions.
(v) Complex Variable (about 10 lectures) :
Introduction to theory of functions of a complex variable.
BOOKS.
(a) Recommended for preliminary reading:
Bell, E. T. Mathematics, Queen and Servant of Science. (McGraw-Iii.) 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:
(ii) Thomas, G. B.--Limits. (Addison-Wesley.) Brand, L.—Advanced Calculus. (Wiley.)
Courant, R. Differential and Integral Calculus. 2 vols. (Blacide. } (altern.) 140
( iii) Munkres, J. P. Elementary Linear Algebra (Addison-Wesley.) Aitken, A. C.—Determinants and Matrices. ( Oliver & Boyd. l
Ayres, F.—Theõrems and Problems of Matrices. (Schaum.) } (altern.) Hohn, F. E.-Elementary Matrix Algebra. ( Macmillan.) J
(iv) Bland, D. R. Solutions of Laplace's Equation. ( Н.К.Р. ) 2 (Meru. ) Sneddon, I. N. Fourier Series. (R.K.P.) j
EXAMINATION Two 3-hour papers
.
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 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.
SYLLABUS
A selection of topics from (i) to ( viii) , together with (ix) and (x) : (i) Elements of mathematical logic.
(ii) Algebra. Introduction to abstract algebra.
( iii) Algebra. Theory of equations.
(iv) Geometry. Projective and non-euclidean geometry
.
(v) Analysis. Convergence. Expansions in infinite series.
(vi) Calculus. Functions of a complex variabl e.
( 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.
Books
( a) Preliminary reading: As for Pure Mathematics part III course A, and also Adler, I.—The New Mathematics. (Mentor.)
Klein, F. Elementarryy Mathematics—Arithmetic, Algebra. Analysis. (Dover.) Meserve, B. E.—Fundamental Concepts of Algebra. ( Addison-Wesley. ) Meserve, B. E.-Fundamental Concepts of Geometry. (Addison-Wesley.) (b) Prescribed textbook: .
Courant, R., and Robbins—What is Mathematics? (0.U.P.) EXAMINATION. Two 3-hour papers.
89. PURE MATHEMATICS PART III-COURSE C
This subject is superseded by the subject Theory of Computation II, for which details will be found in the. Science Handbook. The normal prerequisite to the new subject is of course Theory of Computation I. The Sub-Dean will advise students whose courses are disturbed by these changes.
90. GENERAL MATHEMATICS
A course 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 normally not be accepted as such without further work and the permission of the head of the department of Mathematics.
Students who have passed a Matriculation Mathematics subject, but not in a recent year, are strongly recommended to seek advice as to preparatory work from the Department in January.
Students who have not passed a Matriculation Mathematics subject at all are advised to consult the lecturer before enrolling..
FACULTY OF ARTS HANDBOOK SYLLABUS
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. Curve fitting.
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.
Probability. Probability as degree of belief; probability and frequency. De- velopment and use of the basic probability theorems. Probability and scientific method.
BOOKS
(a) Preliminary reading: As for Pure Mathematics Part I.
(b) Prescribed textbooks:
Cow, Margaret М.—Pure Mathematics. (E.U.P.)
Kaye and Laby—Four-figure Mathematical Tables. (Longmans.) EXAMINATION. , Two 3-hour papers.
94. APPLIED MATHEMATICS PART I
A course of three lectures and one tutorial class per week throughout the year.
SYLLABUS
The principles of applied mathematics, including conservation of mass, momentum and energy.
Vector methods will be used where appropriate.
It will be assumed that students attending this course have passed the subjects 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.
BOOKS
(a) Preliminary reading: At least two of the following:
Kline, М.—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.) von Kannan, T.—Aerodynamics. (McGraw-Hill.) (b) Prescribed textbooks:
Hilton, P. J. Partial Derivatives. (Routledge & Kegan Paul.) Spain, B. Vector Analysis. (Van Nostrand. )
Glauert, M. B. Principles of Dynamics. (R.K.P.)
( с ) Recommended for reference:
Bullen, K. E.—Introduction to the Theory of Mechanics.
( Science Press.) altern.
Synge, J. L. and Griffith, В. A. Principles of Mechanics.
( McGraw-Hill.)
The books by Mach and Darwin from (a)
Prandtl, L.—The Essentials of Fluid Dynamics. (Blackie. ) 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.
SYLLABUS
(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.
BOOKS
(a) For preliminary reading:
Bullen, K. E.—Theory of Mechanics, ch. XIII. (Science Press.)
Weatherburn C. E.—Elementary Vector Analysis. (Bell), and the chapters on partial derivatives and the chain rule for functions of several variables in any standard Calculus book.
or Rutherford, D. E. Vector Methods. ( Oliver & Boyd.) (b) Prescribed textbooks:
Sokolnikoff, I. S. and E. S.—Higher Mathematics for Engineers and
Physicists. ( McGraw-Hill.) alters.
Hildebrand, F. B. Advanced Calculus for Applications.
( Prentice-Hall. )
Synge, J. L., and Griffith, В. A. Principles of Мechanics. (McGraw-Hill.) alt.
Jaeger, J. C. Introduction to Applied Mathematics. ( О.U.Р. ) EXAMINATION. Two 3-hour papers.
96. APPLIED MATHEMATICS PART III
A course of three lectures per week throughout the year, together with practical work.
SYLLABUS
(i) Numerical analysis and computation.
(ii) Elementary hydrodynamics.
( iii) Cartesian 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. (Bell.) BOOKS
Recommended for reference:
Ads, R.-Vectors, Tensors, and the Basic Equations of Fluid Mechanics. (Prentice-Hal.)
-
143
FACULTY
of
ARTS HANDBOOKHartree, D. R. Numeriвal Analysis. ( O.U.P. )
Hildebrand, F. B.—Advanced Calculus for Applications. (Prentice-Hall.) Jeffreys, H» and В. S.=Methods of Mathematical Physics. ( O.U.Р. )
Maxwell, E. A. Coordinate Geometry with Vectors and Tensors. (Oxford.) Sokolnikoff, I. S.-Mathematical Theory of Elasticity. ( McGraw-Hill.) Temple, G.—An Introduction to Fluid Dynamics. ( O.U.Р. )
Weinstock, R.—Calculus of Variations. (McGraw-Hill.)
EXAMINATION. Two 3-hour papers. Before admission to the examination, candid- ates must have satisfactorily completed the prescribed practical work.
HONOURS DEGREE D. SCHOOL OF MATHEMATICS ( For possible combinations with this school see p. 225. )
1. The course for В.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 В.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.
Tutorial classes are held in the earlier years only. Students are expected to do reading and exercises related to the lectures throughout the course, and the work so done each year may be taken into account in the examinations.
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 at least second class honours in Pure Mathe- matics part I and Applied Mathematics part I. In exceptional circumstances students may be admitted without these qualifications; if admitted they will be advised what reading to undertake in the long vacation. 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 in pure or applied mathematics involving the reading and collation of the relevant mathematical literature, and will present a thesis embodying this work. The topic will be chosen, in consultation with the staff of the department, at or before the beginning of the first term, and the thesis will be presented not later than the be- ginning of the third term. The thesis will be taken into account in determining the class list for the final examination.The fourth subject in First Year is to be chosen from any of the groups, but must not be one of Chemistry, Biology or History and Philosophy of Science.
4. The examinations in Pure Mathematics part III and Applied Mathematics part III ( two papers in each ), held at the end of the Third Year, will count as the fust 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 x.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.
8. 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 proceed to the degree of В.Sс. 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,
VACATION READING
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.) Turnbull, H. W.—The Great Mathematicians. (Methuen.) Bell, E.T.—Men
of Mathematics. (Pelican.)Sarton,
G. History
of Mathematics. (Dover.).Hooper, A.-Makers of Mathematics. (Faber.)
van der Waerden, B.
L.—Science
Awakening. (Groningen.) Dantzig,T.—Request
of the Greeks. (Allen & Unwin.)Boyer, C.
В.—History
of the Calculus. (Dover.) ,85. PURE MATHEMATICS PART I See
p. 138.86. PURE MATHEMATICS PART II (Ions)
A course of four lectures per week in the first two terms, and three in the third term, with tutorial work.
This course may be taken by those who have obtained adequate honours in Pure Mathematics I and by those who have passed satisfactorily in Pure lathe.
rnatics II.
SYLLABUS
(i) Numerical Mathematics or Theory of Numbers or an alternative assignment, to be done in the long vacation preceding the course; no lectures given. Intending students should obtain the exercises and instructions from the Mathematics depart- ment in December or January before the course begins. and should band in their work complete not later than 31 March. This work will carry some weight in the examina- tion.
Numerical Mathematics. Exercises on summation of series, difference tables, inter- polation, integration solution of differential equations, curve fitting, simultaneous linear equations and determinants. Calculating machines will be available for this work, on request, and may be used in the Mathematics department.
It will be necessary to obtain, and to read relevant parts of:
Noble,
B. Numerical
Methods, vols. I and II. ( Oliver & Boyd.)Theory of Numbers. Exercises on prime numbers, factorization, congruences, quadratic residues, continued fractions, Diophantine approximation, quadratic forms.
In addition to the exercises and instructions referred to above, it will be necessary to obtain and read:
Davenport,