In the first part of this module we build on multivariate calculus studied in the first year and extend it to the calculus of scalar and vector functions of several variables. Line, surface and volume integrals are considered and a number of theorems involving these integrals (named after Gauss, Stokes and Green) will be discussed. In particular Green’s theorem, which gives a formula for the line integral of a vector field in the plane round a closed curve, is closely related to complex integration considered in the second part of the module. The integral theorems are also useful in many branches of Applied Mathematics and to describe physical quantities that vary in space and in time. For example, this module is a pre-requisite for MATH2044, Fields and fluids, where these methods are used to describe the behaviour of fluids and of electromagnetic fields.
In the second part of this module, we extend our investigation of calculus to functions of a complex variable, once again building on the material studied in the first year. This theory has both great aesthetic appeal and a large number of applications. We focus here on the integration of these functions, particularly along curves in the complex plane. We develop the basic theory and ideas of the integration of a function of a complex variable, use the main theorems such as Cauchy’s theorem and the Cauchy integral formula, and explore some of their consequences, such as the Fundamental Theorem of Algebra and the evaluation of real integrals.
Pre-requisites: (MATH1048 AND MATH1056) OR (MATH1048 AND MATH1059 AND MATH1060) OR (MATH1006 AND MATH1007)
Aims and Objectives
Having successfully completed this module you will be able to:
- Evaluate real and complex integrals using the Cauchy integral formula and the residue theorem.
- Calculate grad, divergence and curl in Cartesian and other simple coordinate systems, and establish identities connecting these quantities;
- Parametrize a path and evaluate some complex integrals directly;
- Use Gauss’s and Stokes’ theorems to simplify calculations of integrals and prove simple results;
- Evaluate line, surface and volume integrals in simple coordinate systems;
Knowledge and Understanding
Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:
- Discover how calculus of real functions can be extended to complex functions, thus developing a completely new perspective on many seemingly unrelated aspects of mathematics covered in other modules, e.g complex Fourier series for solving partial differential equations or stability of approximation methods in numerical analysis.
- Develop an understanding of how calculus, in both its differential and integral aspects, can be seamlessly extended from scalar functions of one variable to vector functions of many variables.
PART A: VECTOR CALCULUS
Review of basic vector algebra
Differentiation of vector valued functions of one variable
Parametrization of curves and line integrals
Parametrization of surfaces and flux integrals
Gradient, divergence and curl; basic theory and simple examples in Cartesian coordinates
Identities in vector calculus
Integral theorems; Gauss, Stokes and Green’s theorems
Scalar potential, vector potential
General orthogonal curvilinear coordinates
PART B: COMPLEX VARIABLE THEORY
Review of analytic functions and Cauchy-Riemann equations
Paths in the complex plane, parametrization, contours
Cauchy integral formula
Liouville’s theorem and fundamental theorem of algebra
Laurent and Taylor series
Singularities, residues and the residue theorem
Evaluation of real integrals
Learning and Teaching
Teaching and learning methods
Lectures, problem classes, private study
|Total study time||150|
Resources & Reading list
Brown J W & Churchill R V. Complex Variables and Applications.
Matthews PC. Vector Calculus.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External