Vector Calculus - MAT00012I

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  • Department: Mathematics
  • Module co-ordinator: Dr. Chris Wood
  • Credit value: 20 credits
  • Credit level: I
  • Academic year of delivery: 2019-20
    • See module specification for other years: 2016-17

Module will run

Occurrence Teaching cycle
A Spring Term 2019-20 to Summer Term 2019-20

Module aims

Part A:

  • To deepen and extend the Stage 1 Calculus module, with a more mature look at the fundamental concepts of infinitesimal calculus from the viewpoint of vector-valued functions of many variables.
  • To discuss the three essential ingredients of calculus - continuity, differentiability, and integrabilitybringing out the distinctive flavour of each theory, and describing their inter-relationships and applications.
  • To gain a deeper understanding of the three famous differential operators of classical vector calculus: div, grad and curl.

Part B:

  • To describe the differential calculus of scalar and vector fields in R3.
  • To extend the theory of integration to volume integrals and integrals over surfaces in R3.
  • To describe the theorems of Stokes and Gauss which link these topics together, and thereby to complete the chain of generalisations of the Fundamental Theorem of Calculus encountered previously.
  • To determine when a potential function and vector potential exist, and calculate them.

Module learning outcomes

At the end of the module you should be able to

Part A:

  • Appreciate the concept of continuity for functions of many variables, and recognize a good supply of continuous functions.
  • Understand the concept of linear approximation, recognize a wide range of differentiable functions of many variables, and differentiate them.
  • Appreciate, and use, the chain rule.
  • Integrate vector and scalar fields along contours in n-dimensional spaces.
  • Discern when a multi-dimensional vector field is conservative, and construct a potential.
  • Find a harmonic conjugate for a harmonic function in the plane.
  • Calculate the flux of a vector field through a planar domain.
  • Express a line integral as a double integral, and vice versa.

Part B:

  • Compute integrals over a variety of regions of space.
  • Parametrise a variety of surfaces in space and compute surface integrals of vector fields.
  • Use index notation to manipulate vector expressions.
  • Work with the fundamental differential operators of vector calculus.
  • Understand the relationship between volume integrals and surface integrals, and surface integrals and line integrals.
  • Use the integral theorems to move from one type of integral to another.
  • Apply the techniques of vector calculus to a range of problems from geometry and physics.
  • Determine when a potential function and a vector potential exist, and calculate them.


Task Length % of module mark
University - closed examination
Closed Examination
3 hours 100

Special assessment rules



Task Length % of module mark
University - closed examination
Closed Examination
3 hours 100

Module feedback

Information is currently unavailable.

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.

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.