To give an introduction to the techniques of vector and tensor calculus and to the study and solution of the partial differential equations which arise in engineering.

After completing this module, the students should be familiar with the use of grad, div and curl and also the Kronecker delta and Levi-Civita tensor.

They should understand the formulation and applications of Stokes'' Theorem and the Divergence Theorem.

They should be able to classify second-order partial differential equations and solve some standard examples in simple situations.

Introduction to vector calculus: grad, div, curl and their use in differential geometry and in continuum mechanics, Line, surface and volume integrals, Stokes theorem, Divergence theorem and their applications to aerodynamics and fluid dynamics (e.g. circulation, vorticity)

Classification of 2nd order PDEs: elliptic, hyperbolic, parabolic.

Boundary conditions (Dirichlet, Neumann); uniqueness of solutions.

Important examples: wave equation, heat equation, Laplace equation. Polar co-ordinates. PDEs for fluid motion discussed using grad, div and curl.

Solutions eg using separation of variables, Fourier series or Fourier/Laplace transforms. Heat diffusion along a rod; vibrations of rectangular memb ranes.