A course of three lectures and one tutorial class per week throughout the year.
Allocation to lecture groups will be listed on the notice boards of the Mathematics Department in the week preceding first term.
158
MATHEMATICs SYLLABUS
(i_) Algebra and Geometry. Vector algebra. Analytical geometry in space.
Sets and groups. Elementary matrices. Determinants. Complex numbers.
(ii) Calculus. Differentiation and integration, with the usual applications.
Sketching graphs. The standard elementary functions. Functions of two variables.
Infinite series. Differential equations.
(iii) Additional topics may be given, selected from: analytical plane geometry, conic sections, introduction to real numbers.
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 Syllabus ), at the Matriculation Examination.
BOOКS
(a) Preliminary reading: One or more of the following:
Room, T. G., and Mack, J. 1.—The Sorting Process. (Sydney U.P. ) Savу er, W. W.—Mathematician's Delight. (Pelican.)
Titchmarsh, E. C,—Mathematics for the General Reader. ( Hutchinson.) Dantzig, Т.—Number, The Language of Science. (Anchor.)
Northrop, E. P.—Riddles in Mathematics. (Pelican.) Hooper, A.—Makers of Mathematics. ( Faber. )
Turnbull, H. W.—The Great Mathematicians. ( Methuen.) ( b ) Prescribed textbooks:
Thomas, C. B.—Calculus and Analytic Geometry. ( Addison-Wesley. )
Fisher, R. C., and Ziebuhr, A. D.—Calculus and Analytical Geometry. ( 2nd ed., Prentice-Hai.) І altern.*
Courant, R., and John, F. Introduction to Calculus and Analysis.
(Wiley.)
Maxwell, E. A.—Algebraic.Structure and Matrices. (C.U.P.)
A book of mathematical tables. ( Кауе and Laby, Four-figure Mathematical Tables (Longmans) will be provided in examinations.)
(c) Recommended for reference:
Ferrer, W. L. Higher Algebra for Schools. (Oxford.)
McArthur, N. and Keith, A. Intermediate Algebra. ( Methuen.) Cow, Margaret, M. Pure Mathematics. (English Universities Press.) Bowran, A. P.—A Boolean Algebra. (Macmillan.)
Dinkines, Flora—Abstract Mathematical Systems. (Appleton-Century.) Weatherbu, C. E. Elementary Vector Analysis. (Bell.) rn
Hummel, J. A.—Vectors. (Addison-Wesley, Paperback.)
(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.)
Apostol, T. М.—Calculus, Vol. I. (2nd ed., Int. Stud. Ed., Blaisdell.) EXAMINATION. Two 3-hour papers.
• Each lecturer will advise on the text be prefers.
384
-
2. PURE MATHEMATICS PART IL PASSA 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 Mathe- matics Department in the week preceding first term.
SYLLABUS
Complex functions. Exponential and related functions of a complex variable.
Differential equations. Standard types of ordinary differential equation of the first and second orders. Singularities of first order equations.
Integrals. Infinite and improper integrals. Reduction formulae. Multiple inte- grals. Curvilinear and surface integrals.
Functions of several variables. Analytical solid geometry. Directional derivative, 159
FACULTY OF ARTS ØBOOY
tangent plane, stationary points, Lagrange multipliers. Change of variables.
Linear algebra. Linear transformations. Matrix algebra. Introduction to eigen values and eigenvectors.
Convergence. Limits of functions. Positive term series, comparison and ratio tests. Absolute and conditional convergence. Power series. Approximations by series.
Series solution of differential equations.
Mappings. Real plane to real plane; Jacobian, with applications. DifTerentiable functions of a complex variable.
Sets and Groups. Elementary theory.
BOOKS
(a) Preliminary reading: at least two of the following:
Sawyer, W. W.—Prelude to. Mathematics. (Pelican.)
Courant, R., and Robbins, H. E.—What is Mathematics? ( 0.ІУ.Р. )
Pol
ya,
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.) Courant, R.—Differential and Integral Calculus.Vols.
I and II. (Blackie.) Chisholm,J.
and Morris, R.—Mathematical Methods for Physics,vol.
II. (North-Holland.)
*Brand, L.—Advanced Calculus. (Wiley.)
*Fulks, W.—Advanced Calculus. (Wiley.)
• Recommended only for those intending to do Pure Matbematics 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 has the form described below in ( i ) ; that for Course B consists of essay assignments.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), (u), (iii ), (iv), (v) listed below.
Course B consists of units (iv ), (v ), (vi ), ( vii) , and one unit selected from ( viii ), (ix) and (x ); together with two essays and vacation reading, as prescribed in lectures.
Units (iv) and (v) will if possible be given in separate lecture groups to the two courses. Units (viii ), (ix) and (x) may not all be available in the one year.
Instruction sheets for the vacation task for Course A and for the essay assign- rents for Course B 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 (vi), (viii) , (ix) or (x) in place of
(iii).
Candidates who do sufficiently well in Course A may, if they apply, be admitted to Pure Mathematics III Honours Course 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 than31
March. This work will carry some weight in the examina- tion.160
( alУera. )
МА ЕМА тгs
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 ready relevant parts of:
Noble, B.—Numerical Methods, vols. I and II. ( Oliver & Boyd.)
Theory of Numbers. Exercises on prime numbers, factorization, congruences,
quadratic residues, co
пtinиебfractions, Diophantine approxhnation, quadratic forms.
In addition to the exercises and instructions referred to above, it will be necessary to obtain and mad:
Davenport, H.—The Nigher Arithmetic. (Hutchinson.)
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 (viii) as an examinable part of this subject.
(ii) Analysis (a ) ( about 25 lectures) :
Convergence of sequences. Series; absolute and conditional convergence. Multi- plication of serie'.
Continuous and differentiate a functions of one real variable. Convergence of infinite ind improper integrals.
Continuous functions of several variables; implicit functions. Functions defined by integrals. Multiple integrals.
(ш ) Analysis (b) (about 15 lectures): `
Uniform convergence of series of functions, with applications to power series.
Fourier series.
(iv) Linear Algebra ( about 20 lectures )
Mappings and linear transformations. Vector spaces. Matrix algebra. Characteristic polynomial; eigenvalues.
(v) Functions of a Complex Variable (about 15 lectures):
Differentiability. Conformal mapping. Contour integration. Residues.
(vi) Noi-EuclideanCeoinetrii ( about
15 lectures) :
Projective geometry. Conics. Hyperbolic and Riemannian geometries.
(vii) Abstract Algebra (about 15 lectures) :
Groups. Rings. Lattices.
(viii) Theory of Numbers (about 15 lectures) :
Factorization. Congruences. Diophantine equations.
(ix) Special Functions. (about 15 lectures) :
Boundary value problems with linear partial differential equations. Legendre polynomials. Bessel functions.
(x) Statistics (about 15 lectures):. .
Theory of probability. Statistical distributions. . BOOKS.
(a) Rпcoтпт enдед for preliminary reading:
Bell, E.
T.-Mathematics,
Queen and Servant of Sciвncв. (McGraw-Hill.) Кasner, E., and Newman, J. R.—Mathematics and the Imagination. (Bell.)Potya, G.—.Induction and Analogy in Mathematics. ( Princeton.) .
Рolya, G. Patterns of Plausible Inference. (Princeton.)Sawyer, W. W: A Concrete Approach to Abstract Algebra. (Freeman.) Adler, I.—The New Mathematics. (Mentor.)
Meserve, B. E.—Fundamental Concepts of Algebra: (Addison-Wesley.) Meserve, B. E.—Fundamental Concepts of Geometry. (Addison-Wesley.) Struik, D. J.—Concise History of Mathematics. (Dover.)
Boyer, C.
В. Bistory of the Calculus. (Dover.) Dantzig, T. Bequest of the Greeks. (Allen & Unwin.) Van der Waerden, B. L.—Science Awakening. (Groningen.) (b) Prescribed textbooks:
(1) and (iii) Thomas, G. B.—Limits. (Addison-Wesley.) Brand, L.—Advanced Calculus. (Wiley.)
Courant, R.—Differential and Integral Calculus. 2 vols. (Blackie.
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В1FACULTY OF ARTS HANDBOOK