Aims & Learning Objectives:
Aims: The first part of the course provides an introduction to vector calculus, an essential toolkit in most branches of applied mathematics. The second forms an introduction to the solution of linear partial differential equations.
At the end of this course students will be familiar with the fundamental results of vector calculus (Gauss' theorem, Stokes' theorem) and will be able to carry out line, surface and volume integrals in general curvilinear coordinates. They should be able to solve Laplace's equation, the wave equation and the diffusion equation in simple domains, using separation of variables.
Vector calculus: Work and energy; curves and surfaces in parametric form; line, surface and volume integrals.
Grad, div and curl; divergence and Stokes' theorems; curvilinear coordinates; scalar potential.
Fourier series: Formal introduction to Fourier series, statement of Fourier convergence theorem; Fourier cosine and sine series.
Partial differential equations: classification of linear second order PDEs; Laplace's equation in 2D, in rectangular and circular domains; diffusion equation and wave equation in one space dimension; solution by separation of variables.