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 introduce the basic concepts which enable one to investigate the calculus of scalar and vector functions of several variables. Line, surface and volume integrals are then considered and a number of theorems involving these integrals (named after Gauss, Stokes and Green) will be discussed. In particular Stokes’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 many if not all physical quantities, which normally vary both in space and in time. For example, this module is a pre-requisite for MATH2044, Applications of Vector Calculus, where these methods are used to describe the behaviour of fluids and electromagnetic fields. In the second part of this module, we extend our investigation of the calculus of a function to complex variables once again building on the material studied in the first year. 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. This theory has both great aesthetic appeal and a large number of applications; we will highlight some of these applications related to the vector calculus developed in the first part of the module.

Aims and Objectives

Module Aims

This module aims to provide the students with some of the essential mathematical tools to model and understand the world around us. Vector calculus is essential to describe real and virtual objects that move in space and time. Complex variables are often the natural setting for many seemingly unrelated problems, from numerical methods to integral transforms and the study of the response of dynamical systems.

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 Study96
Total study time150

Resources & Reading list

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

Matthews PC. Vector Calculus. 



MethodPercentage contribution
Class Test 20%
Exam  (2 hours) 80%


MethodPercentage contribution
Exam %

Repeat Information

Repeat type: Internal & External

Linked modules

(MATH1048 AND MATH1052 AND MATH1056) OR (MATH1006 AND MATH1007) OR (MATH1050 AND MATH1051 AND MATH1052 AND MATH1048) One of the pre-requisites for MATH2044, MATH3071, MATH3072 and MATH6094


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

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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