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The University of Southampton

MATH2045 Vector Calculus and Complex Variable

Module Overview

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.

Aims and Objectives

Learning Outcomes

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

  • 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.
  • 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.
Learning Outcomes

Having successfully completed this module you will be able to:

  • Calculate grad, divergence and curl in Cartesian and other simple coordinate systems, and establish identities connecting these quantities;
  • Evaluate line, surface and volume integrals in simple coordinate systems;
  • Use Gauss’s and Stokes’ theorems to simplify calculations of integrals and prove simple results;
  • Parametrize a path and evaluate some complex integrals directly;
  • Evaluate real and complex integrals using the Cauchy integral formula and the residue theorem.


PART A: VECTOR CALCULUS Review of basic vector algebra Differentiation of vector valued functions of one variable Vector fields 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 Contour integrals Cauchy’s theorem 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

Independent Study90
Total study time150

Resources & Reading list

Matthews PC. Vector Calculus. 

Brown J W & Churchill R V. Complex Variables and Applications. 



MethodPercentage contribution
Coursework 40%
Written assessment 60%


MethodPercentage contribution
Written assessment 100%

Repeat Information

Repeat type: Internal & External

Linked modules

Pre-requisites: (MATH1048 AND MATH1056) OR (MATH1048 AND MATH1059 AND MATH1060) OR (MATH1006 AND MATH1007)


Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Books and Stationery equipment

Recommended texts for this module may be available in limited supply in the University Library and students may wish to purchase reading texts as appropriate.

Course texts are provided by the library and there are no additional compulsory costs associated with the module

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at

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