DEPARTMENT OF MATHEMATICS

MATØMATTCS SYLLABUS

(i) Algebra and Geometry. Sets and groups. Vector algebra. Anаlytiсal geo- metry in space. Elementary matrices. Determinants. Complex numbers.

(ii) Calculus. Differentiation and integration, with the usual applications.

Sketching graphs. The standard elementary functions. Functions of two variables.

Iяßnite series. Differential equations.

(iii) Additional topics may be given, selected from: analytical plane geometry, conic sections, introduction to real numbers, theory of equations.

It will be assumed that students attending this course have passed both the subjects Pure Mathematics and Calculus and Applied Mathematics, or the one subject Pure Mathematics (Alternative S'yllabus), at the Matriculation Examination.

BOOKS

(a) Preliminary reading: One or more of the following:

Room, T. G., and Mack, J. M.—The Sorting Process. (Sydney- U.P.) Sawyer, W. W: Mathematician's Delight. (Pelican.)

Sawwyyer, W. W.—A Path to Modern Mathematics. ( Pelican. )

Titchmarsh, E. C. Hathematics for the General Reader. ( Hutchinson. ) Dantzig, T.—Number, The Language of Science. (Anchor.)

Northrop, E. P.—Riddles in Mathematics. ( Pelican. ) Hooper, A.—Makers of Mathematics. (Faber.) Adler, I. The New Mathematics. (Mentor.) (b) Prescribed textbooks:

Hall, F. 1.—Abstract Algebra. Vol. I. ( C.U.P.) ( alt. ) Maxwell, E. A.—Algebraic Structure and Matrices. ( Cambridge. )

Thomas, G. B.—Calculus and Analytic Geometer

.

( Addison-Wesley.) ( l) * Courant, R., and John, F.

Introduction to Ca

lculus and Analysis. at.

(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. Higher Algebra for Schools. (Oxford.)

Ferrar, W. L. Higher Algebra, the sequel, starting with ch. XV. (Oxford.) Hummel, J. `A. Vectors. ( Addison-Wesley, Paperback.)

Spain, B.—Vector Analysis. ( Van Nostrand.)

Weiss, M. J., and Dubisch, R. Higher Algebra for the Undergraduate. (Wiley.) Cohn, P. М.—Sõlid Geometry. (Routledge. )

Coln, P. M.—Linear Equations. (Routledge.)

bradam., A. F.—Outline Course of Pure Mathematics. (Pergamon.) (d) Students who are particularly interested may also use with profit:

Dwell, C. V.—Advanced Algebra. Vol. I. ( BeI.)

Durell, C. V., and Robson—Advanced Algebra, vol. 2. (Bell.) Durell, C. V., and Robson—Advanced Trigonometry. (Bell.) Apostol, T. M.—Calculus, Vol.

I. ( 2nd

ed., Blaisdell.) EXAMINATION. Two 3-hour papers.

384 - 2. PURE MATHEMATICS PART II. PASS

A course of three lectures per week with practice classes throughout the year.

Allocation to lecture groups will be listed on the notice boards of the lathe- matics Department in the week preceding first term.

sYLLAВUs

Complex functions. Exponential and related functions of a complex variable.

Differential equations. Standard types of ordinary differential equation of the first and second orders.

Integrals. Infinite and improper integrals. Reduction formulae. Multiple inte- grals. Curvilinear and surface integrals.

Functions of several variables. Analytical geometry in space. Directional deriva- tive, tangent plane, stationary points, Lagrange multipliers. Change of variables.

Mappings. Jacobians. Differentiable functions of a complex variable.

Linear algebra. Vector spaces. Linear transformations. Matrix algebra. Eigenvalues and eigenvectors. Diagonalization and the indentificаΡtion of quadric surfaces.

Convergence. Positive term series, comparison and ratio tests. Absolute and con- ditional convergence. Power series and their use in approximate calculations. Series solution of linear differential equations with variable coefficients.

Algebraic structure. Groups. Rings. Fields.

BOOKS

(a) Preliminary reading: at least two of the following:

Sawyer, W. W.—Prelude to Mathematics. (Pelican.)

Curant, R., and Robbins, H. E.—What is Mathematics? (O.U.P. ) Polya, 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 text books:

Maxwell, E. A.—Algebraic Structure and Matrices. (C.U.P.) And one ^{of the} following:

Maxwell, E. A.—Analytical Calculus, vols. III and IV. (C.U.P.) Thomas, G. B.—Calculus and Analytic Geometry. ( Addison-Wesley. )

Chisholm, J., and Morris, R.—Mathematical Methods for Physics, vol. II ( North- Holland.)

Brand, L.—Advanced Calculus. (Wiley.)

*Finks, W.—Aduanced Calculus. (Wiley.)

• Recommended only for those intending to do Pure Øatbematics Part IIIA.

EXAMINATION. Two 3-hour papers.

385-1. and 385-2. PURE MATHEMATICS PART III PASS Course A and Course B each consist of three lectures per week, with practice classes throughout the year; together with a project. The project for Course A is described below in (i) ; that for course B is described below in (ii) .

Course A is concerned with precise mathematical analysis and with mathematical techniques relevant to the exact sciences. Course B is designed mainly for those intending to teach mathematics subjects in schools; it is also appropriate to those whose interest in mathematics is as an element of general culture rather than as a tool of trade.

Course A consists of units (i), (iii), (iv), (v), (vi) listed below.

Course B consists of units (ii), (v) , (vi) , (vii) , (viii) and (ix) .

Units (v) and (vi) will if possible be given in separate lecture groups to the two courses.

Instruction sheets for the projects for both courses will be available from the Mathematics Department in December, before the courses begin. The work done in these projects will carry weight in the examination.

Students entered for Course A may be permitted, if they apply, to offer one of units ( vii ) or ( viii) in place of (iv.) .

Candidates who do sufficiently well in Course A may, if they apply, be admitted to Pure Mathematics III Honours in a subsequent year.

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 hand in ^their work complete not later than 31 March. This work will carry some weight in the examina- tdon.

Numerical Mathematics. Exercises on summation of series, clifference tables, inteØlation, integration, solution of differential equations, curve fitting, simultaneous linear equations and determinants. Calculating machines will be availabl^e

be r t и.

work, on request, and may be used in the Mathematics Department. It Will ^c nary to obtain, and to read relevant parts of:

160

MATHEMATICS

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, H.—The

Higher Arithmetic. (HutcØson. )

A student who is studying, or who has passed, Theory of Computation part I, must choose the Theory of Numbers assignment.

A student who chooses the Theory of Numbers assignment cannot include topic (ix) as an examinable part of the subject.

(ii) Reading, written work and an essay on prescribed mathematical topics.

Details will be given on the instruction sheet for course B, and students are strongly advised to obtain this sheet early so that they can do some of the reading before lectures start.

( iii)

Analysis (

a) ( about 25 lectures) :

Convergence of sequences and series. Multiplication of series. Double series.

Infinite products.

Continuous and diűerentiabie functions of one real variable. Convergence of infinite and improper integrals.

Continuous functions of several variables; implicit functions. Functions defined by integrals.

(iv) Analysis (b)

(about 15 lectures):

Uniform convergence of infinite series and infinite integrals. Applications to power series. Fourier series.

(v) Linear Algebra ( about

15 lectures ):

Vector spaces. Linear transformations. Matrix algebra.

(vi) Functions of a Complex Variable

(about 15 lectures):

Differentiability. Conformal mapping. Contour integration. Residues.

(vii) Non-Euclidean Geometry ( about

15 lectures) :

Projective geometry. Conics. Hyperbolic and Riemannian geometries.

( viii)

Abstract Algebra ( about

15 lectures) : Group. Rings. Fields Lattices.

(ix) 'Theory of Numbers

(about 10 lectures):

Factorization. Congruences. Diophantine equations.

Boons.

(a) Recommended for preliminary reading:

Bell, E.

T.-Mathematics, Queen and Servant of Science. ( McGraw-Hill. )

Кasner, E., and Newman, J.

R.—Mathematics and the Imagination. ( Bell.)

Polya,

G.—Patterns of Plausible Inference.

(Princeton.)

Sawyer, W. W. A

Concrete Approach to Abstract Algebra. ( Freeman.

Meserve, В.

E.—Fundamental Concepts of Algebra. (Addison-Wesley.)

Meserve, B.

E.-Fundamental Concepts of Geometry.

(Addison-Wesley.) (b) Prescribed textbooks:

( iii) and (iv) Thomas, G.

B.—Limiis.

(Addison-Wesley.)

Brand,

L.—Advanced Calculus. (Wiley.) (

Courant,

R.—Di

ff

erential

and

Integral Calculus.

2 vols. (Blackie.) } alum.) (v) Munkres. J.

P.—Elementary Linear Algebra.

(Addison-Wesley.) Ayres,

F.—Theorems and Problems of Matrices. (Sehaum.)

(altem. )

Hohn, F.

E. Elementary Matrix Algebra. ( Macmillan. )

(vi ) to (ix) As advised in lectures.

EXAMINATION Two 3-hour papers.