# Analysis

## Unit aims

The unit aims to provide some basic tools and concepts for mathematics at the undergraduate level, with particular emphasis on fostering students' ability to think clearly and to appreciate the difference between a mathematically correct treatment and one that is merelyheuristic; introducing rigorous mathematical treatments of some fundamental topics in mathematics and preparing students for higher level pure mathematics courses involving analysis.

## Unit description

Analysis introduces the style of logically precise formulation and reasoning that is characteristic of university-level mathematics; it studies the foundations of elementary calculus in this style using logical quantifiers. It starts from basic properties of the real numbers, studies sequences and series, functions and their limit points, and basic results on continuous functions. It also presents a rigourous treatment of differentiation and integration, and includes inverse functions, series, expronential, logarithmic, and trigonometric functions, uniform continuity, and sequences and series of functions.

## Learning objectives

At the end of the unit, the students should:

• be able to distinguish correct from incorrect and sloppy mathematical reasoning, be comfortable with "proofs by delta and epsilon",
• have a clear notion of the concept of limit as it is used in the context of sequences, series and functions,
• have a clear understanding of the basic properties of continuous functions,
• have a clear notion of the concepts of differentiation and integration,
• have a clear understanding of fundamental functions (such as exponential functions),
• have a clear understanding of series,
• have seen proofs of important results in the course and be able to apply these results to solve standard problems.

## Reading and References

Recommended reading:

• C. W. Clark, Elementary Mathematical Analysis, Wadsworth Publishers of Canada, 1982
• J. M. Howie, Real Analysis, Springer-Verlag, 2001.
• S. Krantz, Real Analysis and Foundations, CRC Press, 1991.
• K. A. Ross, Elementary Analysis: The Theory of Calculus, Springer, 2010.

Further reading:

• Bartle, R.G., Sherbert, D.R., Introduction to Real Analysis, John Wiley 2000.

Unit code: MATH10011
Level of study: C/4 (Honours)
Credit points: 20
Teaching block (weeks): 1 and 2 (1-24)
Lecturers: Dr Thomas Jordan and Dr Ivor McGillivray

## Pre-requisites

A in A Level Mathematics or equivalent

None

## Methods of teaching

Lectures, supported by lecture notes with problem sets and model solutions, problems classes and small group tutorials.

## Methods of Assessment

Formative assessment:

• Problem sheets set by the lecturer and marked by the students’ tutors.

Summative assessment:

• Two 1.5h exams (45% each)
• Coursework (10%)

For information resit arrangements please see re-sit page on the intranet.

Please use these links for further information on relative weighting and marking crieteria

Further exam information can be found on the Maths Intranet.