## PH20107: Mathematical methods for physics 2

[Page last updated: 15 October 2020]

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:
• Examination Semester 1 (EX 33%)
• Examination Semester 2 (EX 67%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
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' problem-solving skills and their understanding of mathematical results.

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 infinite-dimensional vector spaces.

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 coordiinate 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 co-ordinates, 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 N-dimensional 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, Gram-Schmidt 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
• USPH-AFB01 : BSc(Hons) Physics (Year 2)
• USPH-AAB02 : BSc(Hons) Physics with Study year abroad (Year 2)
• USPH-AKB02 : BSc(Hons) Physics with Year long work placement (Year 2)
• USPH-AFB10 : BSc(Hons) Physics with Astrophysics (Year 2)
• USPH-AAB10 : BSc(Hons) Physics with Astrophysics with Study year abroad (Year 2)
• USPH-AKB10 : BSc(Hons) Physics with Astrophysics with Year long work placement (Year 2)
• USPH-AFM02 : MPhys(Hons) Physics (Year 2)
• USPH-AFM04 : MPhys(Hons) Physics with Research placement (Year 2)
• USPH-AAM03 : MPhys(Hons) Physics with Study year abroad (Year 2)
• USPH-AAM12 : MPhys(Hons) Physics with Study year abroad (Year 2)
• USPH-AKM03 : MPhys(Hons) Physics with Professional Placement (Year 2)
• USPH-AKM04 : MPhys(Hons) Physics with Professional and Research Placements (Year 2)
• USPH-AFM10 : MPhys(Hons) Physics with Astrophysics (Year 2)
• USPH-AFM11 : MPhys(Hons) Physics with Astrophysics with Research placement (Year 2)
• USPH-AAM10 : MPhys(Hons) Physics with Astrophysics with Study year abroad (Year 2)
• USPH-AKM10 : MPhys(Hons) Physics with Astrophysics with Professional Placement (Year 2)
• USPH-AKM11 : MPhys(Hons) Physics with Astrophysics with Professional and Research Placements (Year 2)

#### PH20107 is Compulsory on the following programmes:

Programmes in Natural Sciences
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Biology with Physics stream) (Year 2)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Biology with Physics stream) with Study year abroad (Year 2)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Biology with Physics stream) with Year long work placement (Year 2)
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Chemistry with Physics stream) (Year 2)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Chemistry with Physics stream) with Study year abroad (Year 2)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Chemistry with Physics stream) with Year long work placement (Year 2)
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Environmental Science with Physics stream) (Year 2)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Environmental Science with Physics stream) with Study year abroad (Year 2)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Environmental Science with Physics stream) with Year long work placement (Year 2)
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Physics with Biology stream) (Year 2)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Physics with Biology stream) with Study year abroad (Year 2)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Physics with Biology stream) with Year long work placement (Year 2)
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Physics with Chemistry stream) (Year 2)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Physics with Chemistry stream) with Study year abroad (Year 2)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Physics with Chemistry stream) with Year long work placement (Year 2)
• UXXX-AFB01 : BSc(Hons) Natural Sciences (Physics with Environmental Science stream) (Year 2)
• UXXX-AAB02 : BSc(Hons) Natural Sciences (Physics with Environmental Science stream) with Study year abroad (Year 2)
• UXXX-AKB02 : BSc(Hons) Natural Sciences (Physics with Environmental Science stream) with Year long work placement (Year 2)
• UXXX-AFM01 : MSci(Hons) Natural Sciences (Biology with Physics stream) (Year 2)
• UXXX-AAM02 : MSci(Hons) Natural Sciences (Biology with Physics stream) with Study year abroad (Year 2)
• UXXX-AKM02 : MSci(Hons) Natural Sciences (Biology with Physics stream) with Professional Placement (Year 2)
• UXXX-AFM01 : MSci(Hons) Natural Sciences (Chemistry with Physics stream) (Year 2)
• UXXX-AAM02 : MSci(Hons) Natural Sciences (Chemistry with Physics stream) with Study year abroad (Year 2)
• UXXX-AKM02 : MSci(Hons) Natural Sciences (Chemistry with Physics stream) with Professional Placement (Year 2)
• UXXX-AFM01 : MSci(Hons) Natural Sciences (Physics with Biology stream) (Year 2)
• UXXX-AAM02 : MSci(Hons) Natural Sciences (Physics with Biology stream) with Study year abroad (Year 2)
• UXXX-AKM02 : MSci(Hons) Natural Sciences (Physics with Biology stream) with Professional Placement (Year 2)
• UXXX-AFM01 : MSci(Hons) Natural Sciences (Physics with Chemistry stream) (Year 2)
• UXXX-AAM02 : MSci(Hons) Natural Sciences (Physics with Chemistry stream) with Study year abroad (Year 2)
• UXXX-AKM02 : MSci(Hons) Natural Sciences (Physics with Chemistry stream) with Professional Placement (Year 2)

 Notes: This unit catalogue is applicable for the 2020/21 academic year only. Students continuing their studies into 2021/22 and beyond should not assume that this unit will be available in future years in the format displayed here for 2020/21. Programmes and units are subject to change in accordance with normal University procedures. Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.Find out more about these and other important University terms and conditions here.