Department of Physics, Unit Catalogue 2009/10 |
PH20020: Mathematics for scientists 4 |
 Credits:
| 6 |
| Level:
| Intermediate |
| Period: | Semester 2 |
| Assessment:
| EX 100% |
| Supplementary Assessment: | PH20020 - Mandatory Extra Work (where allowed by programme regulations) |
| Requisites:
| Before taking this unit you must take PH20019 |
| Description: | Aims: The aim of this unit is to introduce mathematical concepts and techniques required by science students, and to show how these may be used for different applications. It also aims to continue the development of students' problem-solving skills and their understanding of mathematical results. Learning Outcomes: After taking this unit the student should be able to: * evaluate Fourier series and transforms, and use their properties to solve problems; * use transform methods to solve differential equations; * apply Fourier techniques to problems in the physical sciences; * recognise and solve some of the key equations which arise in the natural sciences; * apply the separation of variables method to linear partial differential equations, and solve the resulting ordinary differential equations by series solution. Skills: Numeracy T/F A, Problem Solving T/F A. Content: Fourier series (5 hours): Periodic functions. Harmonic synthesis. Representation as a Fourier series, Fourier components. Expansion of finite range functions. Applications of Fourier series. Complex form of Fourier series and coefficients. Discrete amplitude spectra. Transition to aperiodic functions (7 hours): The Fourier transform. Integral definition and properties of the Fourier transform. Use of tables in evaluating transforms. Solution of differential equations. Dirac delta function. Convolution, sampling theorem. Uses and applications of Fourier techniques in the physical sciences. Linear equations of science (10 hours): Derivation of the diffusion equation as an example of how PDEs arise in nature. Introduction to Laplace's, Poisson and wave equations. Linearity and superposition. Boundary conditions. Solution by separation of variables in Cartesian, cylindrical and spherical coordinate systems. Series solution of ODEs, including Legendre polynomials and Bessel functions. Sturm-Liouville theory. Orthogonality of functions. |