Analysis in Several Variables

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:5
Terms in which taught: Spring / Summer term module
Pre-requisites: MA1FM Foundations of Mathematics and MA1LA Linear Algebra and MA1CA Calculus
Non-modular pre-requisites:
Co-requisites: MA2VC Vector Calculus and MA2RCA Real and Complex Analysis or
Modules excluded:
Module version for: 2016/7

Summary module description:
In this module the concepts of analysis are generalized to a multidimensional context.

Aims:
To revisit familiar notions of analysis, in particular limits and continuity, in terms of analytical and geometrical concepts, and extend them to a more general setting. To define differentiation and integration in a higher dimensional setting.

Assessable learning outcomes:
By the end of the module students are expected to be able to:
?understand the topological basics of analysis and the geometrical nature of the concept of convergence.
?define the notions of continuity and differentiation in a rigorous way for functions of several real variables
?describe critically the difference between total and partial derivative and the practical consequences.
?apply derivatives to estimate local behaviour rigorously

Additional outcomes:
Students should reflect on the concept of locality and local standard representation of differentiable functions.

Outline content:
The fundamental topological notions of distance and neighbourhood are introduced and applied in a classical context. The module deepens the understanding of these concepts and their geometrical meaning. Continuity and differentiation, previously considered in calculus and vector calculus, are defined rigorously. Their geometrical underpinning is explained and given in a precise form that underlines the local nature of these notions. Integration in several variables and Fubini?s theorem are discussed.

Brief description of teaching and learning methods:
Lectures supported by problem sheets and lecture-based tutorials.

Contact hours:
Lectures- 20
Tutorials- 10
Guided independent study- 68
Total hours by term- 98
Total hours for module- 100

Summative Assessment Methods:
Written exam- 80%
Set exercise- 10%
Class test administered by school- 10%

Other information on summative assessment:
One assignment, one class test and one examination.

Formative assessment methods:
Problem sheets

Length of examination:
2 hours

Requirements for a pass:
A mark of 40% overall.

Reassessment arrangements:
One examination paper of 2 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous class test and coursework marks (80% exam, 10% class test, 10% coursework).