# Mathematics

## MTH2008 - Real Analysis (2022)

MODULE TITLE CREDIT VALUE Real Analysis 15 MTH2008 Prof Ana Rodrigues (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
 Number of Students Taking Module (anticipated) 260
DESCRIPTION - summary of the module content

Infinite processes appear naturally in many contexts, from science and engineering to economics. From solving the equation that finds the wave function of a quantum system in physics, processing sensor data in engineering, to calculating prices for options in economics, at the foundation of all of these are infinite processes and the pure mathematics developed to rigorously and correctly handle these processes. That field of pure mathematics is called analysis, and the central object of study in analysis is the limit which further extends to the notions of convergence, continuity, differentiation, and integrability.

In this module, you will be introduced to the pioneering work of Cauchy, Riemann and many other notable mathematicians. By building on material from the first year, we will carefully and rigorously develop notions first in the context of real variables. In particular, we will develop how to rigorously handle real-variable differentiation, Riemann integration, power series, and basic notions of point set topology.

The material in this module is a prerequisite for the study of Complex Analysis (MTH2009), Topology and Metric Spaces (MTH3040), Integral Equations (MTH3042), Fractal Geometry (MTHM004), Functional Analysis (MTHM001), and Advanced Probability (MTHM042). It is recommended for those studying Dynamical Systems and Chaos (MTHM018), and is the basis for applications in economics, science, and engineering.

Pre-requisite modules: MTH1001; MTH1002 (or equivalent)

AIMS - intentions of the module

Analysis is the theory that underpins all continuous mathematics. The objective of this module is to provide you with a logically based introduction to real analysis. The primary objective is to define all the basic concepts clearly and to develop them sufficiently to provide proofs of useful theorems. This enables you to see the reason for studying analysis, and develops the subject to a stage where you can use it in a wide range of applications.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 state and prove key theorems in real analysis using a rigorous approach;
2 develop proofs related to topological concepts such as limits and connectedness;
3 understand the basis of integration of functions of a real variable.

Discipline Specific Skills and Knowledge:
4 apply fundamental mathematical concepts, manipulations and results in analysis;
5 formulate rigorous arguments as part of your mathematical development;

Personal and Key Transferable/ Employment Skills and Knowledge:
6 think analytically and use logical argument and deduction;
7 communicate your ideas effectively in writing and verbally;

8 manage your time and resources effectively.

SYLLABUS PLAN - summary of the structure and academic content of the module

- Topology on R; Bolzano-Weierstrass theorem

- Epsilon-delta function limits; continuity; differentiability in R

- Function classes: C^k, C^infinity etc; Lipschitz continuity

- Review of epsilon-N sequence limits, Cauchy sequences; series of real numbers, sequences and series of functions;

- Formal theory of Riemann integration; integrability of monotonic functions and continuous functions; problems interchanging limits in general

- Continuity and differentiability in R^n, inverse and implicit function theorems.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 38 112 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled Learning and Teaching Activities 33 Lectures including example classes Scheduled Learning and Teaching Activities 5 Tutorials Guided Independent Study 112 Lecture and assessment preparation; wider reading

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
 Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method Exercise sheets 5 x 10 hours All Discussion at tutorials; tutor feedback on submitted answers

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 10 90 0
DETAILS OF SUMMATIVE ASSESSMENT
 Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method Written Exam – closed book 90% 2 hours (January) All Written/verbal on request, SRS Coursework exercises 1 5% 15 hours All Annotated script and written/verbal feedback Coursework exercises 2 5% 15 hours All Annotated script and written/verbal feedback

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
 Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment Written Exam * Written Exam (2hr) (90%) All August Ref/Def Period Coursework exercises 1 * Coursework exercises 1 (10%) All August Ref/Def Period Coursework exercises 2 * Coursework exercises 2 (10%) All August Ref/Def Period

* Please refer to reassessment notes for details on deferral vs. referral reassessment

RE-ASSESSMENT NOTES

Deferrals: Reassessment will be by coursework and/or exam in the deferred element only.  For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only.  As it is a referral, the mark will be capped at 40%

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Web based and Electronic Resources:

ELE: http://vle.exeter.ac.uk

Type Author Title Edition Publisher Year ISBN Search
Set DuChateau, P.C. Advanced Calculus Harper Collins 1992 000-0-064-67139-9 [Library]
Set McGregor, C., Nimmo, J. & Stothers, W. Fundamentals of University Mathematics 2nd Horwood, Chichester 2000 000-1-898-56310-1 [Library]
Set Gaughan, E. Introduction to Analysis 5th Thompson 1998 000-0-534-35177-8 [Library]
Set Burn, R.P. Numbers and Functions: Steps to Analysis Electronic Cambridge University Press 2005 000-0-521-41086-X [Library]
Set Bryant, V. Yet Another Introduction to Analysis Cambridge University Press 1990 978-0521388351 [Library]
Set Abbott, Stephen Understanding Analysis 2nd Springer, New York 2015 [Library]
Set Krantz, Steven G. Real Analysis and Foundations 4th CRC Press, Boca Raton, FL 2017 [Library]
Set Rudin, R. Principles of Mathematical Analysis 3rd McGraw-Hill Book Co. 1976 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH1001, MTH1002
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 5 No Wednesday 26 February 2020 Tuesday 02 August 2022
KEY WORDS SEARCH Analysis; supremum; infimum; series; functions; limits; continuity; differentiability; integrability;