PH20107: Mathematical methods for physics 2
Academic Year:  2019/0 
Owning Department/School:  Department of Physics 
Credits:  12 [equivalent to 24 CATS credits] 
Notional Study Hours:  240 
Level:  Intermediate (FHEQ level 5) 
Period: 

Assessment Summary:  EX 33%, EXTH 67%* 
Assessment Detail: 
*Assessment updated due to Covid19 disruptions 
Supplementary Assessment: 

Requisites: 
Before taking this module you must take PH10007
While taking this module you must take PH20013 OR take PH20060 
Description:  Aims: The aim of this unit is to introduce more advanced mathematical concepts and techniques, and to show how these may be used for different applications. It also aims to continue the development of students' problemsolving skills and their understanding of mathematical results. Learning Outcomes: After taking this unit the student should be able to: * find the eigenvalues and eigenvectors of matrices; * calculate the normal modes of coupled vibrational systems; * calculate and interpret derivatives of vector functions of 1 variable; * parameterise curves; * define and transform between Cartesian, plane polar, cylindrical polar and spherical polar coordinates; * visualise points, lines, planes and volumes in these coordinates; * define scalar, vector and conservative fields; * evaluate and interpret grad, ^{2} and directional derivatives of scalar fields in the above coordinate systems; * evaluate and interpret div and curl of vector fields in the above coordinate systems; * evaluate and interpret line, surface and volume integrals in the above coordinate systems; * identify conservative fields and find their potential functions; * use and interpret vector integral theorems; * 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: Eigenvalues and eigenvectors (6 hours): Revision of matrix algebra. Eigenvalues and eigenvectors of symmetric/Hermitian matrices and their properties. Linear transformations. Normal modes of ball and spring systems. Vector calculus (16 hours): Differentiation of vectors. Space curves; parameterisation of curves, unit tangent vector. Scalar and vector fields in Cartesian coordinates. Gradient and directional derivative of a scalar field, as a vector operator. Div and curl in Cartesian coordinates, physical interpretation. Identities involving , definition of ^{2}. Tangential line integrals. Conservative fields and potential functions. Surface, flux and volume integrals in Cartesian coordinates. Orthogonal curvilinear coordinate systems. Plane polar coordinates; velocity and acceleration, equations of motion. Spherical polar and cylindrical polar coordinates. Line, surface and volume integrals integrals in curvilinear coordinates. Grad, div, curl and ^{2} in curvilinear coordinates; div and curl as limits of integrals. Meaning and uses of ^{2}. Vector integral theorems; Divergence, Stoke's and Green's theorems. Use of vector integral theorems. 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. 
Programme availability: 
PH20107 is a Designated Essential Unit on the following programmes:Department of Physics
PH20107 is Compulsory on the following programmes:Programmes in Natural Sciences

Notes:
