Undergraduate Programme and Module Handbook 2008-2009 (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 2008/09 | Module Cap | None. | Location | Durham |
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#### Prerequisites

- Core Mathematics A (MATH1012) and Core Mathematics B1 (MATH1051) [the latter may be a co-requisite].

#### Corequisites

- Core Mathematics B1 (MATH1012) 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

Subject-specific Knowledge:

- 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.

Subject-specific Skills:

- 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.

Key Skills:

#### 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 for 19 weels and 1 in term 3 | 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 |

three hour written examination | 100% |

#### Formative Assessment:

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