DURHAM UNIVERSITYDepartment of Mathematical Sciences

Level 1 Mathematics modulesCourse Booklet

2012 - 2013

Science LaboratoriesSouth RoadDurham Email: maths.office@durham.ac.ukDH1 3LE Web: www.maths.dur.ac.uk

Contents

1 General Information 3

1.1 Useful Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Course Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Academic progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Durham University Mathematical Society . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7 Booklists and Descriptions of Courses . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7.1 Calculus I – MATH1061 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7.2 Probability I – MATH1061 . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7.3 Linear Algebra I – MATH1071 . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7.4 Analysis I – MATH1051 . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.7.5 Problem Solving I – MATH1041 . . . . . . . . . . . . . . . . . . . . . . . 18

1.7.6 Dynamics I – MATH1041 . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.7.7 Data Analysis, Modelling and Simulation – MATH1711 . . . . . . . . . . 22

1.7.8 Discrete Mathematics – MATH1031 . . . . . . . . . . . . . . . . . . . . . 24

1.7.9 Mathematics for Engineers and Scientists – MATH1551 . . . . . . . . . . 26

1.7.10 Single Mathematics A – MATH1561 . . . . . . . . . . . . . . . . . . . . 28

1.7.11 Single Mathematics B – MATH1571 . . . . . . . . . . . . . . . . . . . . . 30

1.7.12 Statistics – MATH1541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.7.13 Brush Up Your Skills (1H Support Classes) . . . . . . . . . . . . . . . . . 34

1.7.14 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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1 General InformationWelcome to the Department of Mathematical Sciences! 1,200 undergraduates take modules pro-vided by the Department. This booklet provides information on first-year modules offered bythe department. It also contains summary information on key policies related to assessment andacademic progress.

Full details of the department’s policies and procedures are available in the departmental de-gree programme handbooks at http://www.dur.ac.uk/mathematical.sciences/teaching/handbook/, which also contains on on-line version of the course descriptions contained in this booklet.

Information concerning general University regulations, examination procedures etc., are containedin the Faculty Handbooks (www.dur.ac.uk/faculty.handbook) and the University Calendar,which provide the definitive versions of University policy. The Teaching and Learning Hand-book (www.dur.ac.uk/teachingandlearning.handbook) contains information about assess-ment procedures, amongst other things.

You should keep this booklet for future reference. For instance, prospective employers might findit of interest. You can look forward to an enjoyable year.

1.1 Useful Contacts

The first point of contact for issues referring to a particular course or module should be the relevantlecturer. For more general questions or difficulties you are welcome to consult the Course Directoror your Adviser. For queries relating to teaching issues, for example registration, timetable clashes,support for disabilities or illness, you should visit the department to speak to someone in the mainMaths Office (CM201), or send an email to maths.teaching@durham.ac.uk.

Head of Department:maths.head@durham.ac.ukDirector of Undergraduate Studies:Dr Peter Bowcock (CM307, peter.bowcock@durham.ac.uk)

The Course Directors for students are determined by their programme and level of study as fol-lows:Students on Mathematics programmes at level one:maths.1hcoursedirector@durham.ac.ukStudents on Mathematics programmes at level two:maths.2hcoursedirector@durham.ac.ukStudents on Mathematics programmes at levels three and four:maths.34hcoursedirector@durham.ac.ukStudents on Natural Sciences and Combined Honours programmes at all levels:maths.natscidirector@durham.ac.ukStudents on programmes other than Mathematics and Natural Sciences and Combined Honours atall levels:maths.otherprogdirector@durham.ac.uk

We may also wish to contact you! Please keep the Mathematics Office informed of your currentterm-time residential address and e-mail address.

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1.2 Course Information

Term time in Durham is Michaelmas (10 weeks), Epiphany (9 weeks) and Easter (9 weeks). Thereare 22 teaching weeks, and the last seven weeks are dedicated to private revision, examinationsand registration for the subsequent academic year.

Timetables giving details of places and times of your commitments are available on Departmentalweb pages and noticeboards in the first floor corridor of the Department. It is assumed that youread them!You can access your own Maths timetable at www.maths.dur.ac.uk/teaching/ andthen clicking on the ‘My Maths timetable’ link.

Also, teaching staff often send you important information by e-mail to your local ‘@durham.ac.uk’address, and so you should scan your mailbox regularly.

Note that in October it takes time to sort out groups for tutorials and practicals, and so these classesstart in week 2.

1.3 Assessment

Full details of the University procedures for Examinations and Assessment may be found in Section6 of the Learning and Teaching Handbook, http://www.dur.ac.uk/learningandteaching.handbook/ .The Department’s policies and procedures are described in the departmental degree programmehandbook, http://www.dur.ac.uk/mathematical.sciences/teaching/handbook/ . The Departmentfollows the marking guidelines set out by the University Senate:

Degree Class Marking Range(%)I 70−100

II(i) 60−69II(ii) 50 - 59III 40−49

Fail 0−39

Linear Algebra I (MATH1071), Calculus & Probability I (MATH1061) and Analysis I (MATH1051)are assessed by written examination.For Problem Solving & Dynamics I (MATH 1041), 40% of the assessment is based on summativecoursework submitted in the problem-solving part of the module and 60% is based on a writtenexamination on the Dynamics part of the module.For all other first year modules offered by the department, 10% of the assessment is based on sum-mative coursework and 90% is on a written examination. All courses include either summativeor formative assessed work, with assignments being set on a regular basis in lecture-based courses.The purpose of formative and summative assessment of coursework is to provide feedback to youon your progress and to encourage effort all year long.

Regular assignments are marked A-E to the following conventions:

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Grade Equivalent Mark Quality

A ≥ 80% Essentially complete and correct workB 60%—79% Shows understanding,

but contains a small number of errors or gapsC 40%—59% Clear evidence of a serious attempt at the work,

showing some understanding, but with important gapsD 20%—39% Scrappy work, bare evidence of understanding

or significant work omittedE <20% No understanding or little real attempt made

Use of Calculators in Exams The use of electronic calculators is allowed in some module exam-inations and other module assessments. Each student taking modules offered by departments orschools within the science faculty, which specify that calculators be allowed in assessments, willbe offered a calculator, free of charge, at the beginning of their course. The model will be a Casiofx-83 GTPLUS or a Casio fx-85 GTPLUS.

In October 2012 only, students entering level 2, 3 or 4 of their course and who have module assess-ments, which specify that calculators be allowed, will be offered a calculator free of charge. Themodel will be a Casio fx-83 GTPLUS or a Casio fx-85 GTPLUS.

Calculators will become the property of students who will be responsible for their upkeep. Noreplacement calculators will be provided free of charge, but may be available to purchase fromdepartments/schools, depending on availability. The specified calculator will also be generallyavailable, in shops and online, should a replacement purchase be necessary.

Where the use of calculators is allowed in assessments, including examinations, the only modelsthat will be allowed are either a Casio fx-83 GTPLUS or a Casio fx-85 GTPLUS. In particu-lar, examination invigilators will refuse to allow a candidate to use any calculator other than themodel(s) specified, which will be explicitly stated on the front of the examination paper. Duringexaminations no sharing of calculators between candidates will be permitted, nor will calculatorsor replacement batteries be supplied by the Department or the Student Planning and AssessmentOffice.

1.4 Academic progress

The Department is responsible for ensuring that students are coping with the courses and meetingtheir academic commitments.For 1st year modules you are required:- to attend tutorials/ problems classes/ computer practical classes- to sit collections exams- to submit summative or formative assessed work on time to a satisfactory standard.Assessed work which is graded D or E is counted as being of an unsatisfactory standard.Attendance and submission of work is monitored through a database. It is your responsibility toensure that your attendance is recorded by signing the relevant attendance sheets.

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Students who are not keeping up with their commitments will be contacted by course directors tohelp get them back on track.Persistent default will result in a formal written warning, which may be followed by the initiationof Faculty procedures.

Full details of academic progress requirements for the department are available in the departmentaldegree programme handbook, http://www.dur.ac.uk/mathematical.sciences/teaching/handbook/ .

1.5 Durham University Mathematical Society

MathSoc: Necessary and SufficientDurham University Mathematical Society, affectionately known as MathSoc, provides an opportu-nity for maths students (or anyone with an interest in maths) to meet away from lectures.

We arrange a variety of events throughout the year, such as bar crawls, invited speakers, a Christ-mas meal, film nights and the highlight of the year a trip to see Countdown being filmed! Sothere’s something for everyone.

MathSoc also helps the Maths Department to arrange the Undergraduate Colloquia, where depart-mental and external lecturers give talks on their current research. These cover a wide range ofmathematical topics with previous titles including ’Dot-dots, zig-zags and plank-planks’ and ’De-fects of integrable field theory’. These are at a level such that anyone with an interest in maths canenjoy them and they aim to inspire an interest in a part of maths you may not have seen before.

We have our own website (www.durham.ac.uk/mathematical.society), where you will find all themost up-to-date information about the society. Here you will also find our second-hand book list,which has many of the books needed for courses for much cheaper than you will find them in theshops. Last year people saved up to £50 by using this service!

If you would like any more information about either the society itself, or advice on any other aspectof the maths course for example module choices for next year, feel free to get in touch with any ofthe exec listed below or via the society email address (mathematical.society@durham.ac.uk).

To join:

Come and see our stand at the freshers’ fair, or email at any time: it costs only £6 for life member-ship, or £3 for a year.

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1.6 Disclaimer

The information in this booklet is correct at the time of going to press in May 2012. The Univer-sity, however, reserves the right to make changes without notice to regulations, programmes andsyllabuses. The most up-to-date details of all undergraduate modules can be found in the FacultyHandbook on-line at www.dur.ac.uk/faculty.handbook/.

1.7 Booklists and Descriptions of Courses

The following pages contain brief descriptions of the Level 1 courses in Mathematics. The coremodules Linear Algebra I, Calculus & Probability I, Analysis I and Problem Solving & Dynamics Iare compulsory for Mathematics students, and you may also choose to take one or two of the threeoptional modules Data Analysis & Simulation, Discrete Mathematics and Statistics. Supportingthe core modules there is the optional “Brush Up Your Skills” weekly course.

The other three modules offered - Single Mathematics A, Single Mathematics B and Mathematicsfor Engineers and Scientists - are not open to students on Mathematics degrees, but will be ofinterest to Natural Science students or students in other departments who want to take a Level 1Mathematics module. Note that these modules will not allow you to progress to any Level 2 orhigher Mathematics modules.

These descriptions supplement the official descriptions in the module outlines in the faculty hand-book which can be found athttp://www.dur.ac.uk/faculty.handbook/module_search/?search_dept=MATH&search_level=1. Note that the official module outlines contain information on module pre- and co-requisites, excluded combinations, assessment methods and learning outcomes. The descriptionswhich follow supplement this by providing a list of recommended books and a brief syllabus foreach module.

For some modules you are advised to buy a particular book, indicated by an asterisk; for oth-ers a choice of titles is offered or no specific recommendation is given. There are also suggestionsfor preliminary reading and some time spent on this during the summer vacation may well paydividends in the following years.

Syllabuses, timetables, handbooks, exam information, and much more may be found atwww.maths.dur.ac.uk/teaching/, or by following the link ‘teaching’ from the Department’shome page (www.maths.dur.ac.uk). These syllabuses are intended as guides to the modules.The definitive information on course content and expected learning outcomes is in the officialmodule outlines.

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1.7.1 CALCULUS AND PROBABILITY I – MATH1061TERM 1: CALCULUS (36 lectures)

Prof P. M. Sutcliffe

Calculus is elementary mathematics (algebra, geometry, trigonometry) enhanced by the limit pro-cess. Its invention is credited to Isaac Newton and Gottfried Leibnitz in the late seventeenth cen-tury. Leibnitz started his work in 1673, eight years after Newton, but initiated the basic modernnotation for derivative and integral, dx and

R. From 1690 onward, calculus grew rapidly and

reached its present state in roughly a hundred years.This course will seek to consolidate and expand the knowledge you already have of this extremelyimportant area of mathematics. It is designed to be completely accessible to the beginning cal-culus student without sacrificing appropriate mathematical rigour. The underlying emphasis is onthe three basic concepts of calculus: limit, derivative and integral. Applications from the sciences,engineering, business and economics are often used to motivate or illustrate mathematical ideas.This course will be concerned with the nuts and bolts of calculus, while the Analysis I module willrevisit the above concepts and provide a deeper knowledge.Differential equations are introduced in connection with applications to exponential growth anddecay. Many standard ordinary differential equations (ODEs) that appear frequently in applica-tions are first and second order linear differential equations and are solved by methods that takeadvantage of their natural association with the technique of integration.The course will provide numerous exercises.

Recommended Books*Salas, Hille and Etgen, One and several variables calculus, 9th edition, Wiley, 2002 (hardback),ISBN 0471231207.M.L. Boas, Mathematical methods in the physical sciences, Wiley, 1983 (hardback), ISBN0471044091 (paperback is only available second hand: ISBN 0471099600).

Both these books are useful in several modules at level 1 and 2 (Analysis I, Problem Solving andDynamics I, Mathematical Physics II, Analysis in Many Variables II). All mathematicians have tounderstand calculus, so there are many books aimed at this vast market and a wide selection can befound in the University library. A particularly concise book that might appeal to some students is

R. Haggarty, Fundamentals of mathematical analysis, Addison Wesley (2nd edition) ISBN0201631970.

Preliminary Reading: Revise A-level Core Mathematics material in your favourite books.

Calculators

Electronic calculators are not permitted in this examination.

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Outline of course Calculus I

Aim: To master a variety of methods for solving problems and acquire some skill in writing andexplaining mathematical arguments.

Term 1 (30 lectures)

Elementary Functions of a Real Variable: Domain and range. Graphs of elementary functions.Even and odd functions. Exponential, trigonometric and hyperbolic functions. Algebraic combina-tions and composition. Injective, surjective and bijective functions. Theorem of inverse functions.Logarithm function as inverse of exponential function; inverse trigonometric functions.

Limits and Continuity: Informal treatment of limits. Statement of main properties (uniqueness,calculus of limits theorem). Vertical and horizontal asymptotes. Continuity at a point and on inter-vals.

Differentiation : Derivative as slope of tangent line. Differentiability and continuity. Product,quotient and chain rule. Implicit differentiation. Differential equations. Derivative as rate ofchange. Increasing and decreasing functions. Max-min problems.

Integration: Antiderivatives. Fundamental theorem of calculus. Integration by parts and use ofpartial fractions to integrate rational functions. Integration of even/odd functions. Gaussian inte-gration.

Ordinary Differential Equations: First order: separable, exact, homogeneous, linear. Secondand higher order: linear with constant coefficients, importance of boundary conditions, reductionto a set of first order equations, treatment of homogeneous and inhomogeneous equations, partic-ular integral and complementary function.

Taylor’s Theorem: Taylor polynomials. Statement of Taylor’s theorem with Lagrange remainder.Taylor series expansions of ex,sin x,sinh x, log(1+ x).

Functions of several variables: Continuity. Partial differentiation. Chain rule. Taylor polynomialin two variables.

Fourier Series: Orthogonal functions and Fourier series. Convergence, periodic extension, sineand cosine series, half-range expansion. Parseval’s theorem.

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1.7.2 CALCULUS AND PROBABILITY I – MATH1061TERM 2: INTEGRATION & PROBABILITY (26 lectures)

Dr M. C. M. Troffaes.

Multiple sums and multiple integrals appear throughout mathematics and we start with a briefintroduction to the standard methods for evaluating and re-expressing sums and double and tripleintegrals.

Probability is a concept with applications in all numerate disciplines e.g. in mathematics, sci-ence and technology, medicine, engineering, agriculture, economics and many other fields. In thiscourse, the theory of probability is developed with the calculus and analysis available and withapplications in mind. Among the topics covered are: probability axioms, conditional probability,special distributions, random variables, expectations, generating functions, applications of proba-bility, laws of large numbers, central limit theorems.

Recommended Books

The following book is very good:

*M.H. DeGroot & M.J. Schervish, Probability and Statistics, Int’l Edn, Addison-Wesley, ISBN0321500466; £53 (this excellent book is also the recommended text for 2H Statistics and coversboth courses very well).

The DUO site will provide information about some other textbooks.

A lot of information is available from the website en.wikipedia.org/wiki/Probability

Calculators

Electronic calculators are not permitted in this examination.

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Outline of course Probability I

Aim: to develop probabilistic insight and computational skills.

Term 2 (26 lectures)

Multiple Integration: iterated sums, double and triple integrals by repeated integration, volumeenclosed by surface, Jacobians and change of variables.

Introduction to probability: chance experiments, sample spaces, events, assigning probabilities.Probability axioms and interpretations.

Conditional probability: theorem of total probability, Bayes theorem, independent events. Ap-plications of probability.

Random variables: discrete probability distributions and distribution functions, binomial, Pois-son, Poisson approximation to binomial, transformations of random variables. Continuous randomvariables: probability density functions, normal distribution, normal approximation to binomial.

Joint, marginal and conditional distributions.

Expectations: expectation of transformations, variance, covariance, expectations of expectations,Chebyshev’s inequality, weak law of large numbers. Moment-generating functions.

Central-limit theorems.

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1.7.3 LINEAR ALGEBRA I – MATH1071 (56 lectures)

Dr J. Funke / Prof R. S. Ward

Techniques from linear algebra are used in all of mathematics. This course gives an introductionto all the major ideas in the topic. The things you learn in this course will be very useful for mostmodules you take later on.

The first term is concerned with the solution of linear equations and the various ways in whichthe ideas involved can be interpreted including those given by matrix algebra, vector algebra andgeometry. This enables us to determine when a system of equations has a unique solution and givesus a systematic way of finding it. These ideas are then developed further in terms of the theory ofvector spaces and linear transformations. We will discuss examples of linear transformations thatare familiar from geometry and calculus.

Any linear map can be put into a particularly easy form by changing the basis of the space onwhich it acts. The second term begins with the solution of the eigenvalue problem which tells youhow to find this basis. We then go on to generalise the notions of length, distance and angle to anyvector space. These ideas may be used in a surprisingly large range of contexts. We conclude thecourse by showing how all these ideas come together in the applications to geometry and calculusintroduced in the first term.

Recommended Books

• R.B.J.T. Allenby, Linear Algebra, Butterworth-Heinemann, ISBN 0340 610441.

• H. Anton and C. Rorres, Elementary Linear Algebra, Wiley.

• T.S. Blyth, E.F. Robertson, Basic Linear Algebra, Springer, ISBN 1852 336625.

• T.S. Blyth, E.F. Robertson, Further Linear Algebra, Springer, ISBN 1852 334258.

• S. Lipschutz, M.Lipson, Linear Algebra, 4th ed, Schaum’s Outlines, McGraw-Hill, ISBN007154352X.

• D.C. Lay, Linear Algebra and its Applications.

• H. Anton and R.C. Busby, Contemporary Linear Algebra.

• G. Strang, Introduction to Linear Algebra.

Calculators

Electronic calculators are not permitted in this examination.

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Outline of course (continued on next page) Linear Algebra I

Term 1 (30 lectures)

Vectors in Rn (8 lectures)

• vectors, addition and scalar multiplication in Rn with concrete examples in R2 and R3

• scalar product, vector product, triple product

• equations of lines and planes

• examples: scalar and vector equations of lines and planes in R3

• solutions of linear equations as generalisations of lines and planes in R3

Matrices and determinants (8 lectures)

• matrices as mappings in Rn

• examples: dilation, projection, reflection and rotation in R2

• multiplication and inversion of matrices

• determinants and explicit methods for their calculation (row and column expansion)

• examples: areas of parallelograms, volumes of parallelepipeds

• Gauss–Jordan elimination using matrix notation

Vector spaces over R (7 lectures)

• vector spaces and subspaces

• examples: lines and planes in R3

• linear independence, spanning sets, bases and coordinates, dimension

• vector spaces of polynomials

• affine subspaces

Linear mappings (7 lectures)

• definition of linear mapping (examples: projections, reflections, rotations in R3)

• differentiation as a linear mapping (example: polynomials)

• representation of linear mappings by matrices

• change of basis and of coordinates

• composition of linear mappings and matrix multiplication

• kernel, (row and column) rank and image of a linear mapping

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Term 2 (26 lectures plus collection)

Complex numbers and Cn as a vector space (4 lectures)

• complex numbers: addition, multiplication, complex conjugate

• geometric illustration: Argand diagram, de Moivre formula

• complex numbers and roots of polynomials

• Cn as a vector space

Diagonalisation and Jordan normal form (7 lectures)

• eigenvalues and eigenvectors

• explicit calculation with characteristic polynomial

• diagonalisation by change of basis

• Jordan normal forms: invariant subspaces, normal blocks

Inner product spaces (8 lectures)

• Definition and examples: Rn, Cn, polynomials

• Cauchy–Schwarz inequality

• orthonormal bases and Gram–Schmidt procedure

• orthogonal and unitary matrices

• examples: projection, reflections and distances in R2 and R3

• orthogonal complement of a subspace

• diagonalisation of symmetric matrices by orthogonal matrices

Special polynomials (3 lectures)

• linear differential operators

• special polynomials as eigenfunctions

Groups (4 lectures)

• axioms of groups

• examples: GL(n), SL(n), O(n)

• matrix realisation of symmetry groups of polygons

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1.7.4 ANALYSIS I – MATH1051 (37 lectures)

Prof W. J. Zakrzewski

This course deals mainly with ‘limits of infinite processes’. It provides a firm foundation for theoperations of differentiation and integration that you already know something about. In addition,you will learn how to answer questions such as the following:

(a) What is the limit of the sequence (2/1)1,(3/2)2,(4/3)3,(5/4)4, ... of rational numbers? [An-swer: the transcendental number e .]

(b) It is not hard to believe that the geometric series 1 + 1/2 + 1/4 + 1/8 + ... converges to thevalue 2, but what does the series 1+1/2+1/3+1/4+ ... converge to? [Answer: it does notconverge.]

(c) What is the value of the integralR

∞

0x5/2

1+x3 dx? [Answer: it does not exist.]

We shall discuss techniques for answering questions of this sort. But analysis consists of more thansimply problem-solving. Ultimately, it is about constructing logical arguments (proofs), using thecorrect language and style, and what mathematicians call rigour. Acquiring this skill is more im-portant than learning problem-solving tricks, but also more difficult, especially at first. We hopethat by the end of the year, you will be able to invent and write out simple proofs.

Recommended Books

The course material is covered in many books on calculus or analysis that you will find in thevarious libraries. The book by Salas et al, recommended for several other modules, also coversmost of the material in this course.

The following are standard American blockbusters, which also cover material in several other first-year courses:S.L. Salas, E. Hille & G.J. Etgen, One and several variables calculus, 9th edition 2003, Wiley,ISBN 0471231207; £36.95.G.B. Thomas & R.L. Finney, Calculus and Analytic Geometry, 8th (or higher) edition, 1992,Addison-Wesley, ISBN 020160700XR.T. Smith & R.B. Minton, Calculus, McGraw-Hill, 2000, ISBN 007230474XR.L. Finney, M.D. Weir & F.R. Giordano, Thomas’s Calculus, 10th (or higher) edition, 2001,Addison-Wesley-Longman, ISBN 0201441411R.A. Adams, Calculus, 4th (or higher) edition, 1995, Addison-Wesley, ISBN 020150944X

The following are smaller and more specialised English-style books:R. Haggarty, Fundamentals of Mathematical Analysis, Addison-Wesley, second edition, 1993,ISBN 0201631970; £33R. Maude, Mathematical Analysis, Edward Arnold, 1986C. Clark, Elementary Mathematical Analysis, Wadsworth, 1982

Calculators

Electronic calculators are not permitted in this examination.

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Outline of course Analysis I

Term 1 (20 lectures)

Aim: An understanding of the real and complex number systems, an introduction to methods ofanalysis.

IntroductionNumbers: Introduction: the need for a better understanding of real (and complex) numbers. Thenumber systems Z,Q,R (not axiomatics). |x|< c ⇐⇒ −c < x < c, |a|+ |b| ≥ |a+b| ≥ | |a|− |b| |for real (and complex) numbers.Sup and inf: Q,R and the completeness axiom. Sup and inf of subsets of R and of real valuedfunctions. Relation to maxima/minima. sup f + supg≥ sup( f +g)≥ sup f + infg .Limits of Sequences: e,N definition. Basic theorems (uniqueness of limits, COLT, pinching the-orem). (NB Similar theorems for functions will already have been stated in the Calculus module).Bounded monotonic sequence tends to a limit. Bolzano-Weierstrass theorem (bounded sequencescontain a convergent subsequence).Convergence of Series: Infinite series; convergence, examples including ∑n−a. Comparison test,absolute convergence theorem, ratio test, alternating sign test, conditional convergence. Conver-gence and absolute convergence of complex sequences and series.

Term 2 (17 lectures)

Aim: To construct calculus rigorously, to further develop methods of analysis.

Limits and Continuity: Functions of real and complex variables. Epsilon-delta definition of limitof a function. Proof of one or more of basic theorems on limits (sums, pinching theorem etc). Limitof a function as x tends to infinity, limx→∞ xa/ex, limx→∞ logx/xa. Continuity and equivalence withf (limxn) = lim( f (xn)). Sum, composite of continuous functions is continuous. Intermediate Valuetheorem and applications. Bisection proof of max-min theorem.Differentiability: Definition. Differentiability implies continuity. Proof of product rule of differ-entiation. Proof of Rolle’s theorem, Mean Value theorem and applications (NB. Some of theseapplications will already have been covered in the Calculus module).Integration: Brief discussion of Riemann sums if necessary (already mentioned in the Calculusmodule). Fundamental theorem of calculus. |

Rf | ≤

R| f | for real and complex valued f . Con-

vergence ofR

∞

0 f (x)dx, comparison test, absolute convergence theorem, examples. Convergenceof integrals with bounded range but unbounded integrand, comparison test, absolute convergence,examples. State formula for differentiation under the integral sign. (Integral test for convergenceof series).Real and Complex Power Series: Radius of convergence, term-by-term differentiation and inte-gration with examples to show these results are not necessarily true for general (pointwise con-vergent) series of functions. Taylor series (NB Taylor’s theorem has already been covered in theCalculus module).

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1.7.5 PROBLEM SOLVING AND DYNAMICS I – MATH1041TERM 1: PROBLEM SOLVING (20 lectures & seminars)

Dr N. Peyerimhoff

This module gives you the opportunity to engage in mathematical problem solving and to developproblem solving skills. You will work both individually and in groups on a variety of mathematicalproblems.

General aims

You will become familiar with the structure of written mathematics and with fundamental solutiontechniques. You will develop skills for rigorous logical deduction. There will be many opportu-nities to discuss your solutions of particular problems, and to write mathematics accurately andeffectively, and to reflect on the problems and on correct and wrong solutions.

Topics/scope

This module provides an underpinning for subsequent mathematical modules. It should provideyou with the confidence to tackle unfamiliar problems, think through solutions and present rigorousand convincing arguments for your conjectures. The main focus of this course is the developmentof problem solving skills and not the introduction of a substantial amount of new mathematicalcontent. The skills developed should have wide ranging applicability.

Recommended Books

Much of what you will do is based on the following highly recommendable book.

• Kevin Houston, How to think like a mathematician, Cambridge University Press 2009 ISBN9780521719780.

You may find the following other books/sources on the topic also useful.

• George Polya, How to solve it, Penguin 1990 ISBN 9870140124996

• Alan F. Beardon, Creative Mathematics, Cambridge University Press 2009ISBN 9780521130592

• Martin Day, An Introduction to Proofs and the Mathematical Vernacular, downloadable bookat http://www.math.vt.edu/people/day/ProofsBook/

• John Mason, Leone Burton and Kaye Stacey, Thinking Mathematically, Addison Wesley1985 ISBN 0201102382.

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Outline of course Problem Solving

Aim: To enable students to develop their problem solving skills and to think like a mathematician.

Term 1 (lectures & seminars)

Read and understand mathematics: Structure of written mathematics (meaning of definitions,theorems, corollaries, conjectures). Special mathematical vobabulary (examples: if and only if,necessary/sufficient condition, for all, there exists...)

Analyse problems: Formulate a problem clearly and precisely in your own words. Explain it toothers. What is given/what are the hypotheses? What has to be achieved/what is the conclusion?

Think logically and solve problems: Become familiar with the problem (consider easy and ex-treme cases, draw pictures). What type of problem is it? Which methods of proofs are appropriate?Formulate and solve easier/specialised versions of the problem. Explain to others how far you gotwith your solution. Using particular proof techniques.

Write mathematics: Accurate presentation of the problem. Are definitions needed? What aresuitable definitions? Give the objects appropriate/useful names. Can the problem be illustratedby examples/pictures? How to organise/structure the solution? Breaking the solution into smallerpieces. Using logical arguments.

Reflection: Is the proof correct? Are there gaps in the arguments? Are particular cases not cov-ered? Were all hypotheses of the problem used? Find counterexamples when certain hypotheses ofthe problem are omitted. What particular tricks were needed? Does the solution have a key idea?Are there similar/related problems? Do they have similar solutions? Can the problem/method begeneralised? Can the solution be improved/simplified?

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1.7.6 PROBLEM SOLVING AND DYNAMICS I – MATH1041TERM 2: DYNAMICS (17 lectures)

Prof P. M. Sutcliffe

Dynamics concerns evolution with time. In this course we study a model of time-developmentcalled ‘classical mechanics’. This applies to the world around us and describes the motion of ev-eryday objects via ‘forces’. It was invented by Isaac Newton in the 17th century, when it stimulatedrevolutions in astronomy, physics and mathematics. Today it is a cornerstone of applied science.

This introductory course treats firstly the motion of point particles, and then the motion of a certainextended body - a flexible stretched string. Highlights include conservation laws and use of Fourierseries.

We use what you have covered in Calculus I (ordinary and partial differential equations) and LinearAlgebra I (vectors). It is vital to be familiar with this material!

The Dynamics course leads on naturally to the second-year courses ‘Mathematical Physics II’ and‘Analysis in Many Variables II’.

Recommended Books

M. R. Spiegel, Schaum’s Outline of Theory and Problems of Theoretical Mechanics, McGraw-Hill 1967, ISBN 0070602328

There are many other textbooks on Mechanics in the Library at shelfmarks 531, 531.1, 531.2,531.3. eg. French & Ebison, Introduction to Classical Mechanics.

For vibrating strings and Fourier series, use the books recommended for partial differential equa-tions in Calculus I (especially Boas) or else consult the relevant chapter in almost any book on‘Mathematics for Physical Scientists’ (or Engineers). These are at Library shelfmarks 51:53, 51:54,51:62.

Calculators

Electronic calculators are not permitted in this examination.

20

Outline of course Dynamics

Aim: to provide an introduction to Newton Mechanics applied to simple physical systems.

Term 2 (17 lectures)

Frames of reference, reminder of Newton’s laws in vector form: forces, mass, momentum, gravi-tational force, Lorentz force.

Examples of work, energy, angular momentum.

Simple motions: SHO, Pendulum. Oscillations about stable equilibrium. Projectiles. Chargedparticles in constant electromagnetic fields.

Two-body system: central orbits, energy, angular momentum, planetary motion.

Waves and strings: derivation of wave equation for small amplitude oscillations, solution by sepa-ration of variables.

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1.7.7 DATA ANALYSIS, MODELLING AND SIMULATION –MATH1711 (41 lectures)

Dr D. A. Wooff / Dr D. Evans

Term 1 - Data Analysis: Lectures for the first term of the module coincide with the Statisticsmodule (MATH1541) and provide an introduction to data analysis. The topics to be covered are:sources of data, descriptive statistics, exploration of relationships between two or more variables,and a selection of more advanced techniques.

Term 2 - Modelling and Simulation: The second term deals with problems arising in deter-ministic modelling, allowing us to predict the behaviour of physical systems (or to learn that thebehaviour is unpredictable). For instance biological systems modelling populations with diseaseswhich also experience birth and death.

Computers will be used for some demonstrations and for practical classes. The software will be Rfor Windows and Maple for Windows.

There are two lectures and an average of 1.5 hours of computing practicals and problems classesper week. Weekly problems may be taken from the exercise sheets and for practicals. There willbe a Collection examination in January. All these form an integral part of the module.

Recommended Books

Purchase of a book is not necessary. However, background reading is strongly recommended.Much of the material covered in first term lectures may be found in [1]. Many other introduc-tory statistics texts cover most of the basic techniques addressed. Note that various formulae andmethods may differ slightly from book to book, and from lecture material to books. The latter tworeferences cover material for the second term. Other books may be recommended when appropri-ate.

[1] D.S. Moore and G.P. McCabe, Introduction to the practice of Statistics. The latest editionis the 6th edition, W.H.Freeman, 2008, ISBN 978-1429216227; £45. However, the earlier editions(3rd, 4th, 5th) are all good choices and can be purchased online much more cheaply.

[2] L. Edelstein-Keshet, Mathematical models in Biology, First edition, McGraw (ISBN 0075549506)

[3] M.E. Davis and C.H. Edwards, Elementary Mathematical Modelling, Prentice Hall

Calculators

Approved electronic calculators are allowed in the examinations.

22

Outline of course Data Analysis, Modelling and Simulation

Aim: The module is a first course in practical data analysis and computer modelling. The emphasisof the module is upon the understanding of real-life statistical and mathematical problems, anddevelops the basic concepts and methods by example.

Term 1 (20 lectures)

Sources of Data: Controlled experiments. Randomisation. Observational studies. Ethical prac-tice.

Descriptive Statistics: Displaying distributions: stem and leaf plots, histograms. Notation; sum-mation formulae. Describing and summarising distributions: location (mode, mean, median, per-centiles); spread (variance, inter-quartile range); boxplots. Standardisation. Measurements anderrors: outliers (link from boxplots), bias, randomness, chance errors, (informally) central ten-dency. Normal curve; areas under Normal curve; assessing Normality. Misleading graphs.

Exploring Two-Variable Relationships: Graphical representations; scatterplots; visualising tabu-lated data. Assessing association: correlation and covariance. Exploring association: least squaresand linear regression. Residuals, homoscedasticity, root mean square error and prediction. Theregression effect. Association is not causation.

Methods for More than Two Variables: Least squares and multiple regression; two way tables,mean polish and median polish.

Data Analysis Topics: Chosen from the following: non-linear least squares, smoothing, transfor-mations, design of experiments.

Terms 2 & 3 (21 lectures) Smoothing Data: (2): Least-squares, solving normal equations fromfirst principles with data errors.

Discrete Models (8): Populations with birth, death, competition. Difference equations and theirsolution by the z-transform. Logistic equation. Stability and Chaos.

Continuous Models (6): Chemical reactions, continuous population problems, first-order ordinarydifferential equations, Euler’s method, mechanical models (2nd-order systems), phase portraits,equilibria, stability, phase paths and isoclines.

Stochastic Models (5): Random walks and Monte-Carlo quadratures, problems like Buffon’s nee-dle.

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1.7.8 DISCRETE MATHEMATICS – MATH1031 (41 lectures)

Dr A. R. Wade / Dr S. K. Darwin

This module introduces a wide variety of topics, all of them about things which are discrete (likethe integers) rather than continuous (like the real numbers). We will often ask ‘how many?’ ;these counting problems can be simple to state, using ordinary language, but surprisingly difficultto solve, needing both careful common sense and some specific techniques. The second term ofthe course is mostly about graphs. These are not the familiar graphs of functions, but networks -like for example railway lines and stations.

Many of the problems you will tackle cannot be done by any standard method, so you must learnto explain your thinking clearly, in some suitable combination of words, symbols and diagrams.Of course this skill will be very useful for other modules, and the rest of your life.

Discrete Maths has some of its origins in mathematical puzzles and games, but now finds manyand varied applications, usually in setting up structure or organising something. It is fundamentalto computer science.

There are two lectures and one problems class per week. Problems are set weekly to be handed inand there is a compulsory examination (Collections) in January to see how you are going on. InMay/June there is a 3-hour written examination.

Recommended Books

There is no required text, but any of these might be helpful or interesting.

Grimaldi is perhaps the most comprehensive.

Tucker also covers most of the material.

Graham, Knuth and Patashnik is a mine of interesting information and examples, written in a verychatty style.

Wilson’s book is excellent for the graph theory part of the course and goes well beyond.

Marcus is very good on the non-graph theory parts of the course.

R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley 2nd ed. 1994.R.P. Grimaldi, Discrete and Combinatorial Mathematics, Addison-Wesley fifth ed. ISBN 0321211030.D.A. Marcus, Combinatorics - A Problem Orientated Approach , The Mathematical Associa-tion of America 1998, ISBN 0883857103.R. Tucker, Applied Combinatorics, Wiley 3rd ed. 1995, ISBN 0471110914.R.J. Wilson, Introduction to Graph Theory, Longman 4th ed. 1996.

Calculators

Approved electronic calculators are allowed in the examinations.

24

Outline of course Discrete Mathematics

Aim: To provide students with a range of tools for counting discrete mathematical objects.To provide experience of a range of techniques and algorithms in the context of Graph Theory,many with every day applications.

Term 1 (20 lectures)

Principles of Counting: Arrangements and permutations, selections and combinations, mathemat-ical induction, combinatorial vs arithmetical proof. Pigeonhole principle, inclusion - exclusion.

Recurrence Relations and Generating Functions: Recurrence relations, generating functions,partitions.

Terms 2 & 3 (21 lectures)

Graphs: Basic concepts (paths circuits, connectedness etc.) Euler paths, maze algorithms. Planargraphs, Euler’s theorem, the Platonic graphs. A brief introduction to graph colouring, the SixColour Theorem. Greedy algorithm.

Optimisation Algorithms on Graphs: Trees (relevance to searching data structures, genetics,decision problems). Shortest and longest paths. Spanning trees, travelling salesman and relatedalgorithms. Matching/assignment problems. Latin squares.

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1.7.9 MATHEMATICS FOR ENGINEERS AND SCIENTISTS –MATH1551 (61 lectures)

Dr S. A. Abel / Dr C. Kearton

Note: This module is not available to Mathematics students.

This module is intended to supply the basic mathematical needs for students in Engineering andother sciences.

There will be a short online diagnostic test to be completed during the first week. This test isbased on a wide range of A-level mathematics material. The purpose is to help you brush-up onany material you have forgotten or did not cover in great detail at A-level (as not everyone has thesame mathematical background.) It does not count in any way towards your final mark for thismodule. Note that there are also revision classes during the first two weeks of term where you canpractise problems and ask questions.

There are 3 lectures each week and fortnightly tutorials. The tutorials start in Week 3. Problemswill be set to be handed in each week and there is a Collection examination in December to testyour understanding of the first term material. All these form an integral part of the module, andthe homework is summative, constituting 10% of the final module mark.

Recommended Books

Students should consider buying either the two books by Stroud or the book by Stephenson.

K.A. Stroud with additions by Dexter J. Booth, Engineering Mathematics, Palgrave Macmillan(6th edition paperback), ISBN 978-1-4039-4246-3K.A.Stroud with additions by Dexter J. Booth, Advanced Engineering Mathematics, PalgraveMacmillan (4th edition paperback), ISBN 1-4039-0312-3G.Stephenson, Mathematical Methods for Science Students, Longman.

If you are not too confident about the mathematics module then the books by Stroud will provideyou with much support throughout the module. Students have found these books very helpful inprevious years. You will probably already know some of the material in the first book. Stephensonis a more concise text but should also prove useful for parts of the second year mathematics modulefor Engineering students.

All the contents of the course are covered in e-book for engineers by Pearson, which you will beable to purchase on arrival. You may also like to refer to: (all paperbacks)

A. Croft, R. Davison and M. Hargreaves, Engineering Mathematics, Addison-Wesley.M.R. Spiegel, Advanced Calculus, Schaum.M.R.Spiegel, Vector Analysis, Schaum.Calculators

Electronic calculators are not permitted in this examination.

26

Outline of course Mathematics for Engineers and Scientists

Term 1 (28 lectures)

Elementary Functions (Practical): Their graphs, trigonometric identities and 2D Cartesian geom-etry: To include polynomials, trigonometric functions, inverse trigonometric functions, ex; lnx;x;sin(x+ y), sine and cosine formulae. Line, circle, ellipse, parabola, hyperbola.

Differentiation (Practical): Definition of the derivative of a function as slope of tangent line tograph. Local maxima, minima and stationary points. Differentiation of elementary functions.Rules for differentiation of sums, products, quotients and function of a function.

Integration (Practical): Definition of integration as reverse of differentiation and as area un-der a graph. Integration by partial fractions, substitution and parts. Reduction formula, e.g. forR

sinn x dx.

Complex Numbers: Addition, subtraction, multiplication, division, complex conjugate. Arganddiagram, modulus, argument. Complex exponential, trigonometric and hyperbolic functions. Polarcoordinates. de Moivre’s theorem. Positive integer powers of sinu;cosu in terms of multipleangles.

Differentiation: Limits, continuity and differentiability. L’Hopital’s rule. Leibniz rule. Newton-Raphson method for roots of f (x) = 0. Power series, Taylor’s and MacLaurin’s theorem, andapplications.

Vectors: Addition, subtraction and multiplication by a scalar. Applications in mechanics. Linesand planes. Distance apart of skew lines. Scalar and vector products. Triple scalar product,determinant notation. Moments about point and line. Differentiation with respect to a scalar.Velocity and acceleration.

Terms 2 & 3 (33 lectures)

Partial Differentiation: Functions of several variables. Chain rule. Level curves and surfaces.Gradient of a scalar function. Div and curl. Normal lines and tangent planes to surfaces. Localmaxima, minima, and saddle points.

Linear Algebra: Matrices and determinants, solution of simultaneous linear equations. Gaussianelimination for Ax = b. Gaussian elimination with pivoting. Iterative methods - Jacobi, Gauss-Seidel, SOR. Eigenvalues in matrices.

Ordinary Differential Equations: First order differential equations: separable, homogeneous,exact, linear. Second order linear equations: superposition principle, complementary function andparticular integral for equations with constant coefficients, fitting initial conditions, application tocircuit theory and mechanical vibrations.

27

1.7.10 SINGLE MATHEMATICS A – MATH1561 (62 lectures)

Prof P. E. Dorey / t.b.a.

Note: This module is not available to Mathematics students.

This module follows on from A-level mathematics, although many topics will be covered afresh.There are three lectures and one tutorial per week. Problems are set to be handed in each weekand there is a compulsory examination (Collections) in January. These are all integral parts of themodule.

It is important to do the written work conscientiously throughout the year both to prepare yourselffor the examination and because there is continuous assessment for written work.

The material consists of important basic ideas and techniques in calculus and linear algebra whichhave applications in a huge variety of areas of science and mathematics.

Recommended Books

We will follow the content of the following book (RHB) fairly closely; see the syllabus for chapterreferences. K.F. Riley, M.P. Hobson and S.J. Bence, Mathematical Methods for Physics andEngineering, CUP, 3rd ed. 2006 (ISBN 9780521679718).

Many other books contain the same material, and are worth consulting, for example:G. James, Modern Engineering Mathematics, Prentice-Hall, 3rd ed. 2001 (ISBN 0130183199).E. Kreyszig, Advanced Engineering Mathematics, Wiley, 7th ed. 1993 (ISBN 0471507296)

Calculators

Approved electronic calculators are allowed in the examinations.

28

Outline of course Single Mathematics A

Term 1 (30 lectures)

Diagnostic Test (1)

Elementary Algebra and Basic Functions [Riley Ch. 1] (4): Simple functions and equations,trigonometric identities, coordinate geometry. Binomial expansion, properties of binomial coeffi-cients. Some particular methods of proof.

Integration [Riley Ch. 2] (10): Fundamental theorem of calculus. Natural logarithm; hyper-bolic functions. Basic methods of integration including substitution, integration by parts, partialfractions, reduction formulae. Applications of integration.

Complex Numbers [Riley Ch. 3] (7): Addition, subtraction, multiplication, division, complexconjugate, modulus, argument, polar form. Argand diagram, de Moivre’s theorem, eiθ. Trigono-metric and hyperbolic functions. Roots of unity, solutions of simple equations in terms of complexnumbers, the fundamental theorem of algebra.

Limits and Real Analysis [Riley Ch. 2, 4.7] (8): Real numbers versus rational numbers; limits,continuity, differentiability. Basic methods of differentiation. Utilitarian treatment of the Interme-diate Value Theorem, Rolle’s Theorem, Mean Value theorem. L’Hopital’s rule.

Terms 2 & 3 (33 lectures)

Collections exam (1)

Series and Taylor’s theorem [Riley Ch. 4] (10): Summation of series, convergence of infiniteseries, absolute and conditional convergence. Taylor polynomials, Taylor’s theorem with Lagrangeform of the remainder. Convergence of Taylor series. Applications and simple examples of Taylorseries.

Linear equations and matrices [Riley Ch. 8] (22): Systems of linear equations. Gaussian elim-ination. Vector spaces, linear operators. Matrix algebra, addition and multiplication, identity ma-trix and inverses, transpose of a matrix. Determinants and rules for manipulation. Special typesof square matrix. Eigenvalues and eigenvectors. Diagonalization of matrices. Applications to thesolution of linear ODEs with constant coefficients. Quadratic forms.

29

1.7.11 SINGLE MATHEMATICS B – MATH1571 (62 lectures)

Prof R. Gregory / Dr S. A. Abel / Dr P. Heslop / Dr J. Cumming

Note: This module is not available to Mathematics students.

This module follows on from A-level mathematics, although many topics will be covered afresh.There are three lectures and one tutorial per week. Problems are set to be handed in each weekand there is a compulsory examination (Collections) in January. These are all integral parts of themodule.

In the first term we will discuss vector algebra and some applications to mechanics and geome-try, ordinary differential equations – their classification and solutions, and Fourier analysis – therepresentation of functions as linear superpositions of sines and cosines.

In the second and third terms we cover functions of several variables, partial differential equa-tions, and probability. The ideas of differentiation and integration extended to functions of two ormore variables give rise to partial derivatives and multiple integrals. A partial differential equa-tion expresses a relationship involving a function of two or more variables and some of its partialderivatives. Wave motion is one of the many phenomena described by partial differential equa-tions; an example is vibration of a stretched string, such as a guitar string. The final part of themodule provides an introduction to probability.

Recommended Books

First Term: Ordinary differential equations, vector methods and Fourier analysis can be found inmost books on mathematical methods, for example:

G. James, Modern Engineering Mathematics, Prentice-Hall, 3rd Ed. 2001, (ISBN 0130183199).M.L. Boas, Mathematical Methods in the Physical Sciences, Wiley, 2nd ed. 1983, ISBN 0471099600E. Kreyszig, Advanced Engineering Mathematics, Wiley, 7th ed. 1993, ISBN 0471507296.K.F.Riley, M.P.Hobson, S.J.Bence, Mathematical Methods for Physics and Engineering, CUP,3rd Ed. 2061, (ISBN 9780521679718).

Second and Third Terms: Functions of several variables and partial differential equations are cov-ered in the book by Boas recommended in the first term. The book by James covers partial differ-entiation and Fourier series but not partial differential equations.

The chapter on probability in Riley et al. covers this section of the course. The Schaum outlinebook S. Lipschutz, Probability provides lots of examples on fundamental concepts. An alterna-tive, wonderful, but deeper book, which progresses to a significantly higher level, is W. Feller,Introduction to Probability Theory and its Applications, Vol. I, Wiley.

Calculators

Electronic calculators are not permitted in this examination.

30

Outline of course Single Mathematics B

Term 1 (30 lectures)

Diagnostic Test (1)

Vectors [RHB chapter 7] (9):Scalars and vectors. Bases and components. i, j,k notation. Vector algebra. Multiplication of vec-tors: scalar and vector products and their geometrical meaning, length and orthogonality. Tripleproducts. Applications: equations of lines and planes, distances. Derivatives with respect toscalars: velocity, acceleration forces, moments, angular velocity. Two-dimensional polar coor-dinates, spherical and cylindrical polar coordinates.

Ordinary Differential Equations [RHB chapter 14] (12):General properties. First-order first-degree equations: separable, homogeneous, linear, Bernoullisequations. First-order higher-degree equations. Second-order linear equations with constant coef-ficients. Applications to particle dynamics, using Newtons Laws of Motion.

Fourier Analysis [RHB chapter 12] (8):Periodic functions, orthogonality of trigonometric functions. Dirichlet conditions, Fourier repre-sentation and coefficients. Odd and even functions. Complex form. Parsevals theorem.

Terms 2 & 3 (32 lectures)

Partial differentiation [RHB chapter 5] (9):Functions of several variables, graphs. Partial derivatives, differential, exact & inexact differen-tials. Chain rule, change of variables. Solutions of simple partial differential equations, d’Alembert’ssolution of the wave equation. Taylor expansions, critical points.

Multiple integration [RHB chapter 6] (9):Double integrals, in Cartesian and polar coordinates. Triple integrals and integration in cylindricaland spherical polars. Applications. Change of variables in multiple integrals, Jacobians.

Vector Calculus [RHB chapter 10] (8):Differentiation and integration of vectors. Vector fields. Vector operators (div, grad and curl),combinations of vector operators. General curvilinear coordinates.

Probability [RHB chapter 30] (6):Sample space, probability axioms, conditional probability, random variables, independence, prob-ability distributions (binomial and normal distributions), expectation and variance.

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1.7.12 STATISTICS – MATH1541 (41 lectures)

Dr D. A. Wooff / Dr I. R. Vernon

Statistics attempts to make evaluations concerned with uncertainty and numerical conjectures aboutperplexing questions. The focus of the course is upon the understanding of real-life statistical prob-lems. The first term’s lectures coincide with those for the Data Analysis, Modelling and Simulationmodule (MATH1711), and develop the basic concepts, with an emphasis on using computer pack-ages for exploratory data analysis. In term 2 we address mainly inferential techniques.

No prior statistical knowledge is assumed. Students are required to have an A-level (with grade ’C’at least) in a mathematics subject which may or may not be statistics, or an equivalent qualification.

There are two lectures per week and three other hours (a mixture of tutorials, problems classes,and computer practicals) per fortnight. Problems are set weekly to be handed in for assessment.There will be a Collection examination in January.

Recommended Books

Purchase of a book is not necessary. However, background reading is strongly recommended.

Some of the material covered in first term lectures may be found in [1], and this also provides goodbackground for second term lectures. Many other introductory statistics texts cover most of thebasic techniques addressed. Note that various formulae and methods may differ slightly from bookto book, and from lecture material to books.

[1] D.S. Moore, G.P. McCabe and B. Craig, Introduction to the practice of Statistics. The latestedition is the 6th edition, W.H.Freeman, 2008, ISBN 978-1429216227; £45. However, the earliereditions (3rd, 4th, 5th) are all good choices and can be purchased online much more cheaply.

Calculators

Approved electronic calculators are allowed in the examinations.

32

Outline of course Statistics

Aim: The module is designed to be a first statistics course. The emphasis is upon the understand-ing of real-life statistical problems, and develops the basic concepts and statistical methods byexample.

Term 1 (20 lectures)Sources of data: Controlled experiments. Randomisation. Observational studies. Ethical prac-tice.Descriptive statistics: Displaying distributions: stem and leaf plots, histograms. Notation andsummation formulae. Describing and summarising distributions: location (mean, median, per-centiles); spread (variance, inter-quartile range); boxplots. Standardisation. Measurements anderrors: outliers (link from boxplots), bias, randomness, chance errors, (informally) central ten-dency. Normal curve; areas under Normal curve; assessing Normality. Misleading graphs.Exploring two-variable relationships: Graphical representations; scatterplots; visualising tabu-lated data. Assessing association: correlation and covariance. Exploring association: least squaresand linear regression. Residuals, homoscedasticity, root mean square error and prediction. Theregression effect. Association is not causation. Accuracy of prediction.Methods for more than two variables: Least squares and multiple regression; two way tables,mean polish and median polish.Data analysis topics : Chosen from the following. Non-linear least squares, smoothing, transfor-mations, design of experiments.

Terms 2 & 3 (21 lectures)Probability: Basic ideas for probability, probability axioms. Conditional probability. Indepen-dence. Bayes theorem.Random variables: Discrete and continuous probability distributions. Expectation. Variance.Rules for expectations and variances. Law of large numbers.Introducing inference: Binomial distribution. Random sampling; the sample mean. Distributionof the sample mean. Central limit theorem. Normal approximation to binomial.Introduction to confidence intervals and hypothesis testing: Generating confidence intervals.Basic ideas about hypothesis testing, type I and type II errors. Significance tests. P values. Sensi-ble statistical reporting.Inferences for means of Normally distributed populations: Procedures where the variance isknown. Procedures where the sample size is large. t tests. Matched pairs problems. Comparingtwo population means. Comparing population variances. Comparing several population means(Analysis of variance).Methods for categorical data: Fitting hypothesized frequencies to data. Fitting hypothesizedprobability distributions to data. Chi-square tests of homogeneity. Chi-square tests of indepen-dence.Distribution-free methods: Spearman’s rank correlation coefficient. Mann-Whitney-Wilcoxontest, exact and approximate. Wilcoxon signed rank test, exact and approximate.

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1.7.13 Brush Up Your Skills (1H Support Classes)

Dr P. Heslop

Because of widening access, a broadening A-level syllabus and differences in the syllabuses ofdifferent boards, we facilitate revision and consolidation of the key skills required to embark on amathematics degree through the “Brush Up Your Skills” course. The course covers material thatmost students will have seen at A-level, but as well as revision, the course is intended to cover anygaps there may be in any particular combination of A-level modules.

The course consists of 2 problems classes per week which complement the level 1 core modules.Attendance is not compulsory but is initially advised on the basis of a diagnostic test administeredto all students at the beginning of the first term. The course is voluntary and does not form partof the degree, so students may attend only those sessions that deal with subjects where they feelweak. This facility is intended to help students take control of their own learning, recognize areaof weakness and use the resources available to improve them. It is the first step on the road tobecoming an independent learner.

Recommended Books

Salas, S., Hille, E., Etgen, G., Calculus: One and Several Variables, J. Wiley & Sons, 10th ed.,2007Anton, H., Elementary Linear Algebra, Wiley, 9th ed., 2005DeGroot, M. H., Schervish, M. J., Probability and Statistics, Pearson, 3rd ed., 2003

34

Outline of course Brush Up Your Skills

The Brush Up Your Skills course covers basic pre-calculus topics and broadly follows the LinearAlgebra I and Calculus & Probability I syllabuses; most classes are led by questions posed bythe students or suggested by the 1H lecturers so topics in other 1H courses (e.g. Analysis I andProblem Solving & Dynamics I) are also addressed.

Basics: number systems, basic manipulation, quadratic equations, polynomials, partial fractions,linear and non-linear inequalities, exponents and logarithms, topics in discrete mathematics.

Functions: definition, domain and range, graphs, linear and quadratic functions, composition,inverse, modulus function, hyperbolic functions.

Coordinate Geometry: equations and properties of straight lines, general equation of circle, cen-tre and radius, Cartesian and parametric equations of curves.

Trigonometry: trigonometric functions and identities, inverse trigonometric functions, solution oftrigonometric equations.

Differentiation: definition and properties, interpretation as slope, chain rule, sum, product andquotient rules, simple functions defined implicitly or parametrically, maxima and minima, Taylorand Fourier series, differential equations.

Integration: basic definition, as inverse of differentiation, as area under curve, integration meth-ods, definite integrals, multiple integration.

Vectors: definition, basic properties and operations, magnitude, dot and cross products, vectorialgeometry.

Matrices: definition, basic properties and operations, inverse, determinants.

Probability: permutations and combinations, set theory, Venn diagrams, calculus of probabilities,random variables, discrete and continuous distributions, moments, inequalities, approximations,law of large numbers.

35

1.7.14 MAPLE

Dr K. Peeters

Background:

The Department wants students to work towards being independent learners. It has therefore in-corporated the use computer algebra in its teaching. Computer algebra software makes manymathematical calculations and derivations straightforward. It can help you learning and doingmathematics. It may be used to reduce the tedium of extended calculations, to verify the correct-ness of hand calculation and also for exploration of a new topic. It is an indispensable tool forstudents doing final-year projects as part of their Mathematics degree.

Many computer algebra software systems exist. The department has a adopted Maple as its pre-ferred system:

”Maple is a general purpose computer algebra system, designed to solve mathematical problemsand produce high-quality technical graphics. It is easy to learn, but powerful enough to calculatedifficult integrals in seconds. Maple incorporates a high-level programming language which allowsthe user to define his own procedures; it also has packages of specialized functions which may beloaded to do work in group theory, linear algebra, and statistics, as well as in other fields. It can beused interactively or in batch mode, for teaching or research.” (Centre for Statistical andMathematical Computing, Indiana University)

Content:

There will be an initial one-hour supervised/guided session at the beginning of the year, wherestudents will go through a worksheet showing how computer algebra may be used in the contextof some A-level topics.

During the remainder of the year, from time to time lecturers will demonstrate ways in whichcomputer algebra may be used to check calculations, carry out more difficult computations andgain insight into the material being studied. Lecturers may also set problems to be solved bystudents using computer algebra.

Software availability:

Maple is available on all IT service networked computers. Once registered and in possession of acampus card, students who wish to purchase Maple for their own computers will be able to do so atthe substantially discounted price of £15. In addition, several other free computer algebra systemsare available and help with those can be obtained from the lecturer.

Reference materials:

It is not easy to get much insight into software from purely written materials. There are manybooks about Maple or which use Maple but none is particularly suitable for this level. Some typednotes will be provided at the beginning of the year. Maple itself contains a “New users’ Tour”. Agood starting point for on-line material is Google’s Web Directory Maple section:

directory.google.com/Top/Science/Math/Software/Maple/.

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II

Alg

ebra

II

Mat

hem

atic

al P

hysi

cs II

(1)C

odes

and

Act

uaria

l Mat

hem

atic

s II

or

(2)

Cod

es a

nd G

eom

etric

Top

logy

II

Ope

ratio

ns R

esea

rch

III/IV

(1)P

roba

bilit

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d A

ctua

rial M

athe

mat

ics

II o

r(2

)P

roba

bilit

y an

d G

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etric

Top

olog

y II

Dec

isio

n Th

eory

III

Ana

lysi

s I

Pro

blem

Sol

ving

& D

ynam

ics

I

Ana

lysi

s I a

nd P

robl

em S

olvi

ng &

Dyn

amic

s I

if ta

ken

in Y

ear 2

Con

tinuu

m M

echa

nics

III/I

V20

13-2

014

Foun

datio

ns o

f Phy

sics

I or

Fund

amen

tal P

hysi

cs A

Sol

id li

nes

- : s

how

pre

requ

isite

s.

Dot

ted

lines

...:

show

alte

rnat

ive

sets

of p

rere

quis

ites,

onl

y on

e ne

eded

as

prer

equi

site

whe

re a

ppro

pria

te.

The

num

bers

(1) -

(3) b

efor

e th

e m

odul

e na

mes

, ind

icat

e th

at

thos

e w

ith th

e sa

me

num

ber

are

excl

uded

com

bina

tions

.

Alg

ebra

ic G

eom

etry

III/I

V20

13-2

014

Geo

met

ry II

I/IV

Topo

logy

III

Ana

lysi

s III

/IV20

13-2

014

Dyn

amic

al S

yste

ms

III

Pro

babi

lity

III/IV

Sol

itons

III/I

V

Elli

ptic

Fun

ctio

ns II

I/IV

Mat

hem

atic

al B

iolo

gy II

I

Diff

eren

tial G

eom

etry

III

Gen

eral

Rel

ativ

ity II

I/IV

2013

-201

4

Ele

ctro

mag

netis

m II

I

Qua

ntum

Mec

hani

cs II

I

Par

tial D

iffer

entia

l Equ

atio

ns II

I/IV

Sta

tistic

al M

echa

nics

III/I

V

Sto

chas

tic P

roce

sses

III/I

V20

13-2

014

Sta

tistic

al M

etho

ds II

I

Bay

esia

n S

tatis

tics

III/IV

2013

-201

4

Topi

cs in

Sta

tistic

s III

/IV*

App

roxi

mat

ion

Theo

ry a

nd O

DE

s III

/IV

Rep

rese

ntat

ion

Theo

ry II

I/IV

2013

-201

4

Gal

ois

Theo

ry II

I

Num

ber T

heor

y III

/IV

Mat

hem

atic

al F

inan

ce II

I/IV

Das

hed

lines

--- :

sh

ow m

odul

es w

hich

may

be

pre-

or c

o-re

quis

ites.

Foun

datio

ns o

f Phy

sics

2A

and

2B

At l

east

3 M

aths

mod

ules

in 2

nd y

ear,

at l

east

2 a

t Lev

el 2

(3)C

omm

unic

atin

g M

athe

mat

ics

III

(3)M

athe

mat

ics

Teac

hing

III

1 Le

vel 2

Mat

hem

atic

s m

odul

e or

A

naly

sis

I if t

aken

at l

evel

2

1 L

evel

2 M

aths

mod

ule

&

(Pro

blem

Sol

ving

& D

ynam

ics

I or F

ound

atio

ns o

f Phy

sics

I or

Fund

amen

tal P

hysi

cs A

)

Mod

ule

prer

equi

site

s m

ap

Alg

ebra

ic T

opol

ogy

IV

Rie

man

nian

Geo

met

ry IV

Adv

ance

d Q

uant

um T

heor

y IV

* N

ote:

Sta

ts M

etho

ds II

I is

a co

- or p

re- r

equi

site

for T

opic

s in

Sta

ts II

I/IV

Theo

retic

al P

hysi

cs II

or

Foun

datio

ns o

f Phy

sics

III

Pro

ject

IV

37