MA2ASVNU-Analysis in Several Variables

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:5
Terms in which taught: Spring term module
Pre-requisites: MA0FMNU Foundations of Mathematics and MA0MANU Mathematical Analysis and MA1LANU Linear Algebra and MA1RA1NU Real Analysis 1 and MA1RA2NU Real Analysis II
Non-modular pre-requisites:
Co-requisites: MA2VCNU Vector Calculus
Modules excluded:
Current from: 2021/2

Module Convenor: Dr Chris Daw
Email: chris.daw@reading.ac.uk

Type of module:

Summary module description:

In this module the concepts of analysis are generalized to a multidimensional context. It is 16 weeks long.

The Module lead at NUIST is Dr Jinhui Fang.

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 other topics including 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 loca l behaviour rigorously.


• master the calculations of multiple integrals, line integrals and surface integrals, and the connection between them via the integral theorems of Green, Gauss and Stokes.

Additional outcomes:

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

Outline content:

Week 1Ìý Limit and continuity for functions of several variablesÌý Ìý Ìý Ìý Ìý Ìý ÌýÌý

Week 2Ìý Partial derivatives of a function, geometric interpretation, chain rule, directional derivatives and gradients (points of importance and difficulty: higher-order partial derivatives)

Week 3Ìý The mean value theorem and Taylor's theorem for several variables, extreme value (points of difficulty: calcul ate extreme values)

Week 4Ìý Implicit functions, curves and surfaces in implicit form, geometric applications (e.g. tangent plane and normal line of a surface), conditional extreme valuesÌý Ìý Ìý

Week 5Ìý Parametric integrals (proper and improper)

Week 6Ìý Parametric integrals (points of importance and difficulty: uniform convergency, Euler integration)

Week 7Ìý Multiple integrals (double integrals, concept and property, Fubini’s theorem)

Week 8Ìý Reduction of double integrals to repeated single integrals, (points of importance and difficulty: transformation of double integrals, Polar coordinate transformation)

Week 9Ìý Triple integrals, applications of multiple integrals (area of a surface, mass center)

Week 10 Line integrals (type I and II) and their relations (points of importance and difficulty: calculation of integrals)

Week 11 The integral theo rem of Green, connection between line integrals and double integrals

Week 12 Surface integrals (type I and II), their relation (points of importance and difficulty: calculation of integrals)

Week 13 Orientation of a surface (positive, negative)

Week 14 The integral theorem of Gauss and Stokes,(points of importance and difficulty: proficiency of mastering conversion of integrals using the theorems)

Week 15 Review the relation among all the multiple integrals, line integrals and surface integrals

Week 16 Review for final exam

Brief description of teaching and learning methods:

Lectures supported by problem sheets and lecture-based tutorials.

Contact hours:

Ìý	Autumn	Spring	Summer
Lectures		48
Guided independent study:		52
Ìý	Ìý	Ìý	Ìý
Total hours by term	0	100	0
Ìý	Ìý	Ìý	Ìý
Total hours for module	100

Summative Assessment Methods:

Method	Percentage
Written exam	70
Class test administered by School	30

Summative assessment- Examinations:

2 hours

Summative assessment- Coursework and in-class tests:

One examination and a number of class tests.

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:
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 (70% exam, 30% coursework).

Additional Costs (specified where applicable):

Last updated: 17 November 2021