Undergraduate Programme and Module Handbook 2024-2025

# 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 2024/2025 | Module Cap | Location | Durham |
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#### Prerequisites

- Calculus I (Maths Hons) (MATH1081) or Calculus 1 (MATH1061) and Linear Algebra I (Maths Hons) (MATH1091) or Linear Algebra 1 (MATH1071) and Analysis 1 (MATH1051) [the latter may be 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), Mathematical Methods in Physics (PHYS2611)

#### 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 between multi-dimensional spaces, chain rule, inverse and implicit function theorems, curves, curvature, planar mappings.
- Vector calculus, line and surface integrals and integral theorems, suffix notation.
- Stokes and divergence theorems, conservative field and scalar potential.
- Generalised functions, Dirac delta distribution.
- Hilbert space.
- Sturm-Liouville Theory, Generalised Fourier Series.
- Special functions and orthogonal polynomials.
- Green's functions for ordinary and partial differential equations.
- Method of images for elliptic 2D partial differential equations.

#### 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 Vector Calculus.
- 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.
- The solution of Partial Differential Equations by separation of variables and relation to special functions.
- Sturm-Liouville theory, Fourier and use of Green’s functions to solve ordinary and partial differential equations.

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 | 52 | 2 or 3 lectures per week on an alternating basis throughout Michaelmas and Epiphany terms and two lectures in week 21 | 1 Hour | 52 | |

Tutorials | 10 | Fortnightly for 21 weeks | 1 Hour | 10 | ■ |

Problems Classes | 9 | Fortnightly for 20 weeks | 1 Hour | 9 | |

Preparation and Reading | 129 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

Written examination | 3 hours | 100% | Yes |

#### Formative Assessment:

Weekly or Fortnightly written or electronic assessments.

■ 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