## MA2ASV-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
Modules excluded:
Current from: 2020/1

Module Convenor: Dr Marta De Leon Contreras

Type of module:

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 topics may include 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 d erivatives to estimate local behaviour rigorously.

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 disc ussed.

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

Contact hours:
 Autumn Spring Summer Lectures 20 2 Tutorials 10 Guided independent study: 68 Total hours by term 98 2 Total hours for module 100

Summative Assessment Methods:
 Method Percentage Written exam 80 Set exercise 20

Summative assessment- Examinations:

2 hours.

Summative assessment- Coursework and in-class tests:

Two assignments and one examination.

Formative assessment methods:

Problem sheets.

Penalties for late submission:

The Support Centres will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.
The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment 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, 20%  coursework).