Multivariable and Vector Analysis

This is a year two module.  It is taught intensively by Birmingham academics over eight weeks. There is continuous assessment and a final examination.

Most models of real world situations depend on more than one variable and the techniques of calculus can be extended to solve problems arising in such situations. Typically these are problems whose solutions are functions of position, describing, for example, heat distribution or velocity potential, and involve the partial differentiation or multiple integration of functions of more than one variable. 

The theory and classification of stationary points of functions of two or more variables is developed allowing maxima and minima, including those subject to constraints, to be identified. The differential operators div, grad, curl and the Laplacian are introduced. These are used in particular in the integral theorems (the Divergence theorem and the theorems of Green and Stokes) that relate line, surface and volume integrals and are used in the mathematical formulation of physical conservation laws.

This module develops fundamental ideas that are used both in applied mathematics and in the development of analysis.

By the end of the module students should be able to:

  • use the notation and basic manipulative techniques of the calculus of functions of several real variables
  • apply a variety of analytic and numerical techniques to solve problems in the calculus of several real variables, e.g. to find and analyse the stationary points of functions of more than one variable
  • evaluate line and multiple (surface and volume) integrals
  • evaluate grad, div, curl and the Laplacian in Cartesian and orthogonal curvilinear coordinates
  • State and apply the integral theorems of vector analysis, namely Stokes' and Green’s theorems and the divergence theorem
  • recognise conservative vector fields and their properties