390. PSYCHOLOGY PART IIIB

A course of lectures and experimental work throughout the year.

The lecture course will cover the areas of perception, behaviour theory, cognition, personality, and experimental design.

As well as the practical work contained in the above-named courses, students will carry out two independent projects under supervision and submit reports on these projects. These reports will form part of the annual examination.

Bоoкs. As for Psychology III, together with reading lists provided during the year.

EXAMINATION. Two 3-hour papers together with laboratory notebooks and independent project reports.

85. PURE MATHEMATICS PART I

A course of three lectures and one tutorial class per week throughout the year.

SYLLABUS. (i) Algebra and Geometry. Vector algebra. Analytical geometry in plane and space. Determinants. Sketching graphs. Complex numbers. ,Introduc- tion to sets and groups.

(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 elementary matrices, conic sections, functions of two variables.

It will be assumed that students attending this course have passed the subject Pure Mathematics at the Matriculation Examination.

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

Read, A. Н.—Sigп¢ost 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.)

Courant, R., and Robbins, H.—What is MathematicsP (О.U.Р.) (b) Prescribed text-books:

Thomas, G. B.—Calculus and Analytic Geometry. (Addison- Wesley.)

Purcell, E. J.—Calculus with Analytic Geometry. (Appleton- Century.)

Courant, R., and John, F.—Introduction to Calculus and Analysis. (Wiley.)

(alteri.) *

109

Weatherburn, C. E. Elementarý Vector Analysis. (Bell.) t (altern.) lummel, J. A.—Vectors. (Addison-Wesley.) J

A book of mathematical tables (Kaye and Laby—Four-figure Mathematical Tables (Longmans.) will be provided in examinations.)

( с) Recommended for reference:

Ferrar, W. L.—Higher Algebra for Schools. (Oxford.) McArthur, N., and Keith, A.—Intermediate Algebra. (Methuen.) Gow, Margaret, M.—Pure Mathematics. (E.U.P.)

Bowran, A. P.—A Booleaa Algebra. (Macmillan.)

Dinkines, Flora. Abstract ltfathematical Systems. (Appleton-Century.) Weiss, M. J., and Dubisch, R. Higher Algebra for Undergraduates. (Wiley.)

(d) Students who are aiming at honours may also use with profit:

Ferrar, W. L.—Higher Algebra, the sequel, starting with Ch. 15. (O.U.P.) Duren, C. V., and Robson—Advanced Algebra, Vols. I and II. (Bell.) Duren, C. V., and Robson—Advanced Trigonometry. (Bell.)

ExAmiNATION. Two 3-hour papers.

* Each lecturer will advise on the texts he prefers.

86. PURE MATHEMATICS PART II

HONOURS COURSE

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 Part I, and by those who have passed satisfactorily in Pure Mathematics Part II.

SYLLAВus. (i) Numerical Mathematics or Theory of Numbers or an altern- ative 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 department 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 examination.

Numerical Mathematics. Exercises on summation of series; difference tables, interpolation, integration, solution of differential equations, curve fitting, simul- taneous 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 and 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. (Hutchinson.) (ii) Analysis (about 60 lectures).

Convergence of sequences, Series ; absolute and conditional convergence.

Differentiable and continuous functions of one real variable. Riemann integral.

Convergence of infinite and improper integrals.

Double series, multiplication of series, partial fraction expansions.

Uniform convergence of series of functions. Power series, including the ele- mentary functions of a complex variable. Fourier series. Integral transforms.

Continuous functions of several variables. Differentiability, change of vari- ables, implicit functions. Functions defined by integrals. Multiple integrals.

(iii) Algebra and Geometry (about 30 lectures).

Linear transformations. Vector spaces. Matrix algebra. Characteristic poly- nomial. Quadratic forms. Systems of linear equations.

(iv) Additional topic in Geometry, Algebra or Analysis (at most 10 lectures).

110

Booкs. (a) Recommended for preliminary reading: at least two of Courant, R»

and

Robbins, H.

E.—What is Mathematics?

(O.U.P.) Reid,

C.—Introduction to Higher Mathematics.

(Routledge and Kegan Paul.) Sawyer, W.

W.—Prelude to Mathematus.

(Pelican.)

Bell, E. Т.—Mathematics, Queen and Servant of Science. (McGraw-Hill.) Coxeter, H.

S.—Introduction to Geometry.

(Wiley.)

Hilbert, D.,

and

Cohn-Vossen—Geometry

and the Imagination.

(Chelsea.) Kasner, E.,

and

Newman, J.

R.—Mathematics

and

the Imagination.

(Simon

and Schuster.)

Stab er, E.

R. Introduction to Mathematical Thought.

(Addison-Wesley.) Waismann,

F.—Introdu'tion to Mathematical Thinking.

(Harper Torchbook.)

(b) Prescribed text-books:

Ferrar, W.

L.—Textbook of Convergence.

(O.U.P.

Hyslop, J.

M. Infinite Series.

(Oliver & Boyd.) } (altern.) Brand,

L.-Advanced Calculus.

(Wiley.)

Fucks,

W.—Advanced Calculus.

(Wiley.) (altern.

)

Olmsted, J. M.

H.-Advanced Calculus.

(Appleton.)

Buck, C.

R. Advanced Calculus.

(McGraw-Hill.)

Ferrar, W.

L. Algebra: Determinants, Matrices, etc.

(O.U.P.

1

Murdoch, D.

C.—Linear Algebra for Undergraduates.

(Wiley) } (altern.) Hohn, F.

E. Elementary Matrix Algebra.

(Macmillan.)

J

ExAISINATiox. Two 3-hour papers.

86. PURE MATHEMATICS. PART II.

PASS COURSE

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

SYLLABUS. - -

Complex

functions.

Exponential and related functions of a complex variable.

Di fferential Equations.

Standard types of 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, tangent plane, stationary points. Change of variables. Polar coordinates.

Linear Algebra.

Linear transformations. Matrix algebra. Introduction to eigenvalues and eigenvectors.

Convergence.

Concept of a limit (real and complex). Series; absolute and conditional convergence, comparison and ratio tests. Power series. Approximations by series. Series solution of differential equations.

Mappings.

Real plane to real plane ; Jacobian, with applications. Diffferentiab e functions of a complex variable.

Sets and Groups.

Elementary theory.

Bоокs. (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? (O.U.P.)

Рolуа,

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

One

of the following:

Maxwell, E.

A.—Analytical Calculus,

Vols. III and IV. (C.U.P.) Thomas, G.

1.—Calculus and Analytic Geometry.

(Addison-Wesley) Courant,

R.—Di fferential 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.)

* Recommended only for those intending to do Pure Mathematics Part IIIA.

ExAMiNATiox. Two 3-hour papers.

111

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 Mathematics or Theory of Numbers or an alternative assign- ment, to be done in the long vacation preceding the course; no lectures given.

Intending students should obtain the exercises and instructions from the lather- atics department 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 examination.

Numerical Mathematics. Exercises on summation of series, difference tables, interpolation, integration, solution of differential equations, curve fitting, simul- taneous 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 and Boyd.)

Theory of Numbers. Exercises on prime numbers, factorization, congruences, quadratic residues, continued fractions, Diophartine 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. (Hutchinson.) (ii) Analysis. (about 40 lectures) :

Convergence of sequences. Series; absolute and conditional convergence.

Multiplication of series.

Continuous and differentiable 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. Vector spaces.

(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.

Вooкs. (a) Recommended for preliminary reading:

Bell, E. T.—Mathematics, Queen and Servant of Science. (McGraw-Hill.) Kasncr, E., and Newman, J. R.—Mathematics and the Imagination. (Bell.)

Courant, R., and Robbins, H. E.—What is Mathematics?

(O.U.Р.) (Earlier parti of chapters I, II, VI, VIII.) Sawyer, W. W. Prelude to Mathematics. (Pelican.)

Sawyer, W.

W.

—A Concrete Approach to Abstract Algebra. (Freeman.) (b) Prescribed text-books:

(ii) Thomas, G. B.—Limits. (Addison-Wesley.) Brand, L. Advanced Calculus. (Wiley.)

Courant, R.--Differential and Integral Calculus. (2 vols.) - (altern.) (Вlасkiе)

(iii) Munkres, J. P.—Elementary Linear Algebra. (Addison-Wesley.) Aitken, A. C. Determinants and Matrices. (Oliver & Boyd.)

Ayres, F.—Theorems and Problems of Matrices. (Schaum.) (altern.) Hohn, F. E.—Elementary Matrix Algebra. (Macmillan.)

112

(iv) Bland, D. R.—Solutions of Laplace's Equation. (Rout

-

ledge & Kegan Paul.)

Sneddon, I. N.—Fourier Series. (Routledge & Kegan Paul.) ЕxAmINАТюx. Two 3-hour papers.

88. PURE MATHEMATICS PART III—COURSE

Ё

.

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 subjects ; 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 development, and in another direction gives some indication of the role of mathematics in science, culture and society.

SYLLnвvs. 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 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.

Booxs. (a) Prescribed for preliminary reading:

As for pure Mathematics Part III course A, and also Adler, I.—The New Mathematics. (Mentor.)

Klein, F. Elementary 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 text-books : As advised in lectures ; and Courant, R.. ana! Robbins.—What is Mathematicst (O.U.P.) ExAmiNAІЇox. Two 3-hour papers.

89. PURE MATHEMATICS PART III—COURSE C

This subject is superseded by the subject Theory of Computation II, for which the details will be found on p. 117. The normal pre-requisite on the new subject is of course Theory of Computation I.

The Sub-Dean will advise students whose courses are disturbed by these changes.