PH20107: Mathematical methods for physics 2
[Page last updated: 21 April 2022]
Academic Year:  2022/3 
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 100% 
Assessment Detail: 

Supplementary Assessment: 

Requisites: 
Before taking this module you must take PH10007
While taking this module you must take PH20013 OR take PH20060 
Learning Outcomes:  After taking this unit the student should be able to:
* calculate and interpret derivatives of vector functions of one 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; * use spherical harmonics to represent functions defined on the surface of a sphere; * outline key properties of finite and infinitedimensional vector spaces. 
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. 
Skills:  Numeracy T/F A, Problem Solving T/F A. 
Content:  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. Fourier transforms (7 hours): Integral definition and properties of the Fourier transform. Use of tables in evaluating transforms. Solution of differential equations. Dirac delta function. Convolution, sampling theorem. Discrete and Fast Fourier transforms. Uses and applications of Fourier techniques in the physical sciences. Linear equations of science (13 hours): Linear operators and linear PDEs. Classification of PDEs, examples of linear PDEs in the physical sciences. Superposition, boundary conditions. Solution by separation of variables in Cartesian, cylindrical and spherical coordinate systems. Fourier transform method for solving PDEs. Series solution of ODEs, recurrence relation. Convergence of infinite series solutions, and relevance to PDEs describing a physical situation. Ordinary and singular points. Bessel functions, Legendre polynomials and associated Legendre polynomials. Laplace's equation in spherical polar coordinates, spherical harmonics, and their use for representing functions defined on a sphere. Solving the Schrödinger equation for the hydrogen atom. Matrices and vector spaces (3 hours): Basis vectors, definition of a vector space, complex Ndimensional vector spaces. Inner product, norm. Linear operators acting on vectors. Symmetric, Hermitian, orthogonal and unitary matrices. Eigenvectors and eigenvalues. Commutation. Infinite dimensional vector spaces, Hilbert space. Basis functions, GramSchmidt orthogonalisation. Hermitian operators. Applications to quantum mechanics, Dirac notation revisited. Hermitian operators. Applications to quantum mechanics, Dirac notation revisited. 
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:
