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Vector & Complex Calculus - MAT00047I

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  • Department: Mathematics
  • Module co-ordinator: Dr. Francesco Cosentino
  • Credit value: 20 credits
  • Credit level: I
  • Academic year of delivery: 2023-24
    • See module specification for other years: 2024-25

Module summary

This module encompasses two subjects. Vector calculus is about differentiation and integration of scalar and vector fields in 2 and 3 dimensions. Complex analysis concerns the differentiation and integration of functions of a complex variable.

Related modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

Pre-requisite modules:

  • Foundations & Calculus
  •  Multivariable Calculus & Matrices

Module will run

Occurrence Teaching period
A Semester 1 2023-24

Module aims

This module aims to deepen and extend knowledge from first year Calculus, with a more mature look at the fundamental concepts of infinitesimal calculus from the viewpoint of vector-valued functions of many variables. The three variable case underpins all continuous processes in the three space dimensions of our world, and which is therefore essential to the application of mathematics in the natural sciences.

The two variable case leads to the notion of differentiable functions of a complex variable which have very many properties beyond those of the differentiable functions of real variables that were studied previously. These underpin the important technique of contour integration, which is useful for evaluating real integrals.

Module learning outcomes

By the end of this module, students will be able to:

  1. Solve a variety of vector calculus problems in 2 and 3 dimensions using the Stokes theorem, Gauss’ theorem, and Greens’ theorem.

  2. Apply gradient, divergence, curl using different coordinate systems such as spherical polar, cartesian, and cylindrical polar.

  3. Compute flux and surface integrals in different coordinate systems.

  4. Apply the Cauchy-Riemann equations and discuss when a function is analytic.

  5. Compute the Laurent series and identify the poles of a complex function.

  6. Use the Residue Theorem to compute the value of a definite integral

Module content

In the first part of this module, we discuss the three essential ingredients of calculus—continuity, differentiability, and integrability—bringing out the distinctive flavour of each theory, and describing their inter-relationships and applications; we gain a deeper understanding of the three famous differential operators of classical vector calculus: div, grad and curl; we describe the theorems of Stokes and Gauss which link these topics together.

The second part of this module will extend the ideas developed about real functions to complex functions, develop the theory of holomorphic functions, and apply this theory to understand problems arising in real analysis or calculus. This lays the groundwork for future study of Fourier analysis which is widely used in real-world applications. These techniques are used by many graduates in their day-to-day work.


Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Vector & Complex Calculus
3 hours 100

Special assessment rules


Additional assessment information

There will be five formative written assignments, with marked work returned in the seminars


Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Vector & Complex Calculus
3 hours 100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

KF Riley, MP Hobson, SJ Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide, CUP 2006 (3rd ed)

H F Davis & A D Snider, Vector Analysis, Allyn & Bacon.

H A Priestley "Introduction to Complex Analysis" (2 ed) OUP 2003, JBM S7

A Pinkus and S Zafrany, Fourier Series and Integral Transforms (S 7.39 PIN)

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.