Module MATH2031: ANALYSIS IN MANY VARIABLES II

Department: Mathematical Sciences

MATH2031: ANALYSIS IN MANY VARIABLES II

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), 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 on n-dimensional Euclidean space, open sets, continuity, differentiability, extrema of functions of more than one variable, Lagrange multipliers
• 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.
• Sturm-Liouville Theory, Generalised Fourier Series
• 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.
• Sturm-Liouville theory, Fourier Transforms and the Heat Kernel

• 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

Summative Assessment

Formative Assessment:

Weekly or Fortnightly written 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