Undergraduate Programme and Module Handbook 2014-2015 (archived)
Module MATH2031: ANALYSIS IN MANY VARIABLES II
Department: Mathematical Sciences
MATH2031:
ANALYSIS IN MANY VARIABLES II
Type |
Open |
Level |
2 |
Credits |
20 |
Availability |
Available in 2014/15 |
Module Cap |
None. |
Location |
Durham
|
Prerequisites
- Calculus and Probability 1 (MATH1061) and Linear Algebra 1 (MATH1071) and Analysis 1 (MATH1051) [the latter may be a co-requisite].
Corequisites
- Analysis 1 (MATH1051) unless taken
before.
Excluded Combination of Modules
- Mathematics for Engineers and Scientists (MATH1551), Single
Mathematics A (MATH1561), Single Mathematics B (MATH1571), Foundation
Mathematics (MATH1641)
Aims
- To provide an understanding of calculus in more than one dimension,
together with an understanding of and facility with the methods of
vector calculus.
- To understand the application of these ideas to a range of forms of
integration and to solutions of a range of classical partial
differential equations.
Content
- Functions on n-dimensional Euclidean space, open sets,
continuity, differentiability.
- functions between multi-dimensional spaces, chain rule,
inverse and implicit function theorems, curves, curvature, planar
mappings, conformal mappings.
- Vector calculus and integral theorems, suffix
notation.
- Multiple integration, line, surface and volume integrals,
Stokes and divergence theorems, conservative field and scalar
potential.
- Solution of Laplace and Poisson equations, uniqueness,
Green's functions, solution by separation of variables.
- Fourier transforms and inverse, convolution theorems,
solution to heat equation using Fourier transform and construction of
heat kernel, connection with Green's function.
Learning Outcomes
- By the end of the module students will: be able to solve a
range of predictable and unpredictable problems in Analysis in Many
Variables.
- have an awareness of the abstract concepts of theoretical
mathematics in the field of Analysis in Many Variables.
- have a knowledge and understanding of fundamental theories of
these subjects demonstrated through one or more of the following topic
areas: differential and integral vector calculus.
- the divergence and Stokes' theorems.
- solution of Partial Differential Equations by separation of
variables.
- solution of Ordinary Differential Equations by power series
expansions.
- In addition students will have the ability to undertake and
defend the use of mathematical skills in the following areas with
minimal guidance: Modelling, Spatial awareness.
Modes of Teaching, Learning and Assessment and how these contribute to
the learning outcomes of the module
- Lecturing demonstrates what is required to be learned and the
application of the theory to practical examples.
- Weekly homework problems provide formative assessment to guide
students in the correct development of their knowledge and
skills.
- Tutorials provide active engagement and feedback to the
learning process.
- The end-of-year examination assesses the knowledge acquired
and the ability to solve predictable and unpredictable
problems.
Teaching Methods and Learning Hours
Activity |
Number |
Frequency |
Duration |
Total/Hours |
|
Lectures |
42 |
2 per week |
1 Hour |
42 |
|
Tutorials |
10 |
Fortnightly for 20 weeks |
1 Hour |
10 |
■ |
Problems Classes |
10 |
Fortnightly for 20 weeks |
1 Hour |
10 |
|
Preparation and Reading |
|
|
|
138 |
|
Total |
|
|
|
200 |
|
Summative Assessment
Component: Examination |
Component Weighting: 100% |
Element |
Length / duration |
Element Weighting |
Resit Opportunity |
Written examination |
3 hours |
100% |
Yes |
One written assignment to be handed in every third
lecture in the first 2 terms. Normally each will consist of solving
problems from a Problems Sheet and typically will be about 2 pages long.
Students will have about one week to complete each
assignment.
■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University