The module extends the calculus of several variables (introduced in MA2001) to the description and analysis of vector and scalar fields. There will be an emphasis on ideas and results that can be applied in many areas of mathematical modelling. But the main vector calculus theorems that will be presented are also of significance without regard to any such applications. This is because they can be viewed as being natural extensions – to cases involving more than one-dimension - of the fundamental theorem of calculus which relates the process of integration to that of anti-differentiation.

• Appreciate the utility of various vector calculus conceptions in the description of vector and scalar fields that appear in mathematical models.
• Calculate and interpret various derivatives, line integrals, surface integrals and volume integrals for vector and scalar fields.
• Make use of simple non-Cartesian co-ordinates in the description of vector and scalar fields.
• Understand, verify and apply the main vector calculus theorems presented in the module.

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills:
The capability to analyse and describe a range of properties of vector and scalar fields, including their derivatives and integrals.

Transferable Skills:
Calculation, visualisation and interpretation skills from vector calculus provide essential ingredients for many areas of mathematical modelling, as well as being of fundamental mathematical importance.

Formative assessment is carried out by means of regular tutorial exercises. Feedback to students on their solutions and their progress towards learning outcomes will be provided in lectures.  

Summative assessment takes the form of a coursework exercise and written examination at the end of the module. The summative assessment gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions

• Vector and scalar fields
  • parameterised curves and tangent vectors
  • gradient operator, directional derivatives, divergence, curl and Laplacian
  • vector identities
  • examples of mathematical models involving vector and scalar fields and their derivatives

• Non-Cartesian co-ordinates
  • cylindrical polar co-ordinates
  • spherical polar co-ordinates

• Line integrals
  • application to energy change
  • independence from parameterisation
  • Double integrals: Green’s theorem in the plane (statement and illustration).

• Surfaces
  • parameterisation of surfaces, including examples for the cylinder and sphere
  • tangent vectors
  • normal and unit normal vectors
  • Surface integrals: calculation using parameterisation, Stokes theorem (statement and illustration).
  • Volume integrals: divergence theorem (statement and illustration).

Not applicable.

Advanced Engineering Mathematics,  Kreyszig, E., Wiley.