Vector Calculus module (MA31007)

Vector calculus is a fascinating subject that allows us to understand dynamic behaviour in 3D space. It is a beautiful subject in its own right, with many deep and elegant results. It is also a powerful tool for studying the behaviour and properties of vector fields and their applications in physics and engineering.

A scalar field is a function that assigns a single number to every point in a space. For example, the temperature at any location can be described by a scalar field. In contrast, a vector field is a function that assigns a vector to each point in a space. For example, the velocity of a fluid at any point can be represented by a vector field.

The main operations of vector calculus are:

• the gradient of a scalar field, which measures the rate of change of the field in any direction
• the divergence of a vector field, which measures the net outward flux of the field through a surface
• the curl of a vector field, which measures the net rotation of the field around a point

These operations are related by various technical results, such as the Divergence theorem and Stokes’ theorem, which relate the integrals of vector fields over different types of regions, such as curves, surfaces, and volumes.

What you will learn

In this module, you will:

• explore different systems of coordinates that can be used in 3D
• learn about curves, surfaces, and volumes in 3D space
• be introduced to the differential operators gradient, divergence, and curl
• be introduced to line integrals, surface integrals, and volume integrals
• learn about the Divergence theorem and Stokes' theorem

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

• set up appropriate coordinate systems in 3D space
• carry out calculations involving the gradient of scalar fields, and the divergence and curl of vector fields
• find appropriate parameterisations of curves and surfaces
• calculate integrals along curves and over surfaces or volumes
• apply the Divergence theorem or Stokes' theorem to assist with the calculation of integrals

Assignments / assessment

• coursework (20%)
  • homework exercises
  • short in-class tests
• exam (80%)

Teaching methods / timetable

• weekly lectures
  • key points of the week's content will be discussed
  • lecture notes covering the full module content will be given before classes
  • video content on key concepts will also be available
• weekly tutorials
  • two hours per week
  • solve problems individually and in groups
  • support with difficulties in learning will be provided by your lecturer and peers