Duration: 22 weeks
Number of lectures per week: 2 lectures plus 1 tutorial
Assessment: Weekly tutorial problems.
End-of-year Examination: One 3-hour examination
Description: This course follows on directly from 2E1/2E2 and develops the mathematics of engineering and physics. It covers Fourier series, Fourier transforms, partial differential equations, linear programming and optimisation, complex analysis.
This section is based on Kreysig chapters 10-11.
Fouriers Theorem. Even and Odd Functions. Half-Range Fourier Series.
Derivation of Fourier Transform from complex Fourier Series. Linearity of the Fourier transform and the Fourier tranform of a derivative. Application to partial differential equations.
Partial differential equations. Wave equation with d'Alemberts solution. Method of separation of variables applied to solutions of the diffusion (heat) equation, Laplace's equation and the wave equation subject to appropriate initial and boundary conditions. Fourier series applied to matching initial conditions. Natural modes and nodal lines. Classifications of partial differential equations in two variables.
This section is based on Kreysig Chapter 20 and part of Chapter 21.
It will introduce some basic aspects of linear programming and graph theory.
Complex function and mappings. Complex Differentiation. Analytic functions. Cauchy-Riemann equations theorem and its proof. Harmonic functions. Power series and radius of convergence (without proofs). Exponential, trigonometric and hyperbolic functions (for complex arguments). Logarithm and complex power functions. Mapping properties of some examples.
Complex Integration. Cauchy's integral theorem and its proof. Cauchy's integral formula. Independence of path consequence of Cauchy's theorem and use of the theory to evaluate complex integrals in simple cases (residue theorem not covered). Power series representations in discs.
See http://www.maths.tcd.ie/~richardt/3E1/ for further information.
Erwin Kreyszig, Advanced Engineering Mathematics, (8th edition) Wiley, 1999.
May 16, 2003