Cookies

We use cookies to ensure that we give you the best experience on our website. You can change your cookie settings at any time. Otherwise, we'll assume you're OK to continue.

Durham University

Faculty Handbook Archive

Archive Module Description

This page is for the academic year 2021-22. The current handbook year is 2022-23

Department: Mathematical Sciences

MATH2031: ANALYSIS IN MANY VARIABLES II

Type Open Level 2 Credits 20 Availability Available in 2021/22 Module Cap Location Durham

Prerequisites

  • Calculus 1 (MATH1061) and 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, conformal 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 function
  • Sturm-Liouville Theory, Generalised Fourier Series as a basis for infinite dimensional vector space
  • Separation of variables in circular and spherical problems
  • Frobenius method and Bessel, Legendre and other special functions
  • Fourier Transforms and Green's functions
  • Laplace and other transforms
  • The calculus of variations, the catenary and other applications
  • 2D PDEs and conformal mapping
  • Method of images

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 Laplace transforms and use of Green’s functions
  • Calculus of variations
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