Lecturer: Florian Theil

Term(s): Term 1

THIS MODULE IS NOT AVAILABLE TO MATHS STUDENTS

Commitment: 30 one hour lectures

Assessment: 85% 2 hour examination, 15% Assignments

Formal registration prerequisites: None

Assumed knowledge:

• MA137 Mathematical Analysis - This module finishes one variable calculus started in MA137
• MA106 Linear Algebra - A few ideas from vector spaces and the use of matrices

Useful background: No further background is required. For one application it would be useful to be familiar with ordinary differential equations as studied in school or MA133 Differential Equations or MA113 Differential Equations A

Synergies: Natural modules that accompany this one are:

• MA259 Multivariable Calculus
• MA260 Norms, Metrics and Topologies or MA222 Metric Spaces
• MA269 Asymptotics and Integral Transforms also links well with the ideas of this module. Ideas in the module will be explored in a more general setting in year three in MA359 Measure Theory and ST342 Mathematics of Random Events

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA222 Metric Spaces
• MA250 Introduction to Partial Differential Equations
• MA260 Norms, Metrics and Topologies
• MA209 Variational Principles
• MA3H0 Numerical Analysis and PDEs
• MA3D9 Geometry of Curves and Surfaces
• MA3K0 High Dimensional Probability
• MA3G7 Functional Analysis I
• MA359 Measure Theory
• MA3H7 Control Theory
• MA3G1 Theory of Partial Differential Equations

Content: This covers three topics:

• Integration
• Convergence of sequences and series of functions
• Norms

The idea behind integration is to compute the area under a curve. The fundamental theorem of calculus gives the precise relation between integration and differentiation. However, integration involves taking a limit, and the deeper properties of integration require a precise and careful analysis of this limiting process. This module proves that every continuous function can be integrated, and proves the fundamental theorem of calculus. It also discusses how integration can be applied to define some of the basic functions of analysis and to establish their fundamental properties.

Many functions can be written as limits of sequences of simpler functions (or as sums of series): thus a power series is a limit of polynomials, and a Fourier series is the sum of a trigonometric series with coefficients given by certain integrals. The second part of the module develops methods for deciding when a function defined as the limit of a sequence of other functions is continuous, differentiable, integrable, and for differentiating and integrating this limit. Norms are used at several stages and finally applied to show that a Differential Equation has a solution.

Aims:

• To develop a good working knowledge of the construction of the integral of regulated functions
• To study the continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions
• To use the concept of norm in a vector space to discuss convergence and continuity there

Objectives: By the end of the course the student should understand:

• The need for a rigorous theory of integration, and that this can be developed for regulated functions by approximating the area under the graph by rectangles
• Uniform and pointwise convergence of functions together with properties of the limit function
• Be able to prove the main results of integration: any continuous function can be integrated on a bounded interval and the Fundamental Theorem of Calculus
• Prove and apply the Contraction Mapping Theorem

Books:
There is no book that covers the whole module Analysis III in the style of the lectures. However, it may be useful to consult titles suggested on the module page for Regulated Functions, Uniform Convergence or the Fundamental Theorem of Calculus. Students registered for this module may access the relevant chapters of books scanned under copyright.

Additional Resources