PHYS30672
Mathematical Methods for Physics
10
Semester 2
Undergraduate
Level 3
Alexander Grigorenko
Unit coordinator
Department of Physics & Astronomy
1
FHEQ level (Framework for Higher Education Qualifications) ' Last part of a Bachelors '
European Credit Transfer & Accumulation System Rating : 5.0
<p>The unit covers four main interlinked areas of mathematical physics: Sturm-Liouville Theory, Green’s Functions, Integral Equations and Calculus of Variations. All four sections have a dual focus: on the more formal properties of the equations, including their consequences for, for instance, the completeness of the eigenfunctions of Hermitian operators, but also on solving problems, including those with source terms, that occur in classical and quantum physics. Differential equations, which make the core of modern physics, will be discussed; their Green’s functions will be derived; the correspondence between differential and integral equations will be highlighted; Calculus of Variations will be elucidated.</p>
<p>The unit covers four main interlinked areas of mathematical physics: Sturm-Liouville Theory, Green’s Functions, Integral Equations and Calculus of Variations. All four sections have a dual focus: on the more formal properties of the equations, including their consequences for, for instance, the completeness of the eigenfunctions of Hermitian operators, but also on solving problems, including those with source terms, that occur in classical and quantum physics. Differential equations, which make the core of modern physics, will be discussed; their Green’s functions will be derived; the correspondence between differential and integral equations will be highlighted; Calculus of Variations will be elucidated.</p>
<p>The aim of this course is to achieve an understanding and appreciation, in as integrated a form as possible, of some mathematical techniques which are widely used in theoretical physics. </p>
<p>On completion successful students will be able to:</p><p><span class="cf0">Recognize when a Green's function solution is appropriate and construct the Green's function for some well-known physical equations.</span></p><p><span class="cf0">Recognize and solve particular cases of Fredholm and Volterra integral equations.</span></p><p><span class="cf0">Describe the basic properties of the eigenfunctions of Sturm-Liouville operators; derive the solutions in particular cases.</span></p><p><span class="cf0">Solve a variational problem by constructing an appropriate functional, and solving the Euler-Lagrange equations.</span></p>
<p class="pf0"><span class="cf0">Mathematical Methods for Physics</span></p><p class="pf0"> </p><p class="pf0"><span class="cf0">The unit covers four main interlinked areas of mathematical physics: Sturm-Liouville Theory, Green’s Functions, Integral Equations and Calculus of Variations. All four sections have a dual focus: on the more formal properties of the equations, including their consequences for, for instance, the completeness of the eigenfunctions of Hermitian operators, but also on solving problems, including those with source terms, that occur in classical and quantum physics.</span></p>
<p>Two one hour, live in-person lectures per week where the core material with examples will be delivered. The recordings of these lectures will be on the podcast system. The lectures are accompanied by full notes and summaries online. This is augmented by weekly online short quiz questions with immediate solutions, and fortnightly sheets on in-depth problems, which are discussed in the examples classes. A Piazza discussion forum is also provided where students can ask questions with answers provided by other students and the unit lead. Formative feedback will be provided during example classes.</p>
1
Written exam
100%
<p>Feedback will be available on students’ individual written solutions to examples sheets, which will be marked, and model answers will be issued.</p>
PHYS20171
Mathematics of Waves and Fields
Pre-Requisite
Compulsory
PHYS30201
Mathematical Fundamentals of Quantum Mechanics
Pre-Requisite
Compulsory
PHYS20672
Complex Variables and Vector Spaces
Pre-Requisite
Compulsory
<div class="ewa-rteLine" style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;"><strong>Pre-Requisites:</strong></div><div class="ewa-rteLine" style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;">Complex Variables and Vector Spaces PHYS20672 ;</div><div class="ewa-rteLine" style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;">Lagrangian Dynamics PHYS20402 OR PHYS20401 Lagrangian Dynamics (2025/26 only)</div><div class="ewa-rteLine" style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;"> </div><div class="ewa-rteLine" style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;"><strong>Anti-requisites:</strong></div><div class="ewa-rteLine" style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;">PHYS40672 Mathematical Methods for Physics (2025/26 only)</div><div class="ewa-rteLine" style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;">Not available to Maths/Physics</div>
N
<p>Reading list is given during the lectures.</p>
Assessment written exam
1.5
Lectures
24
Practical classes & workshops
4
0
70.5