Study information

# Mathematics with Physical Applications - 2023 entry

MODULE TITLE CREDIT VALUE Mathematics with Physical Applications 15 PHY2025 Prof Jacopo Bertolotti (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 11
 Number of Students Taking Module (anticipated) 146
DESCRIPTION - summary of the module content

The emphasis in this module is on practical skills rather than formal proofs. Students will acquire skills in some key mathematical techniques that relate directly to the advanced modules they will meet in the later stages of their degree programme, but also have wide applicability across the mathematical sciences.

AIMS - intentions of the module

This module aims to enable the student to build on the knowledge and skills developed in PHY1026 in order to achieve a deeper understanding of and greater competence in some central mathematical ideas and techniques used throughout physics.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)
A student who has passed this module should be able to:

Module Specific Skills and Knowledge:
1. use probability theory to solve problems;
2. calculate Fourier transforms and use them to solve problems
3. solve partial differential equations by separation of variables;
4. calculate eignvalues and eigenvectors and apply the the techniques to physical problems;
5. use basis vectors to transform differential operator equations to matrix form and hence apply eigen equation techniques;
6. obtain approximate solutions to differential equations through the use of perturbation theory;
7. solve problems involving classical particles by applying the Lagrangian formulation classical mechanics;
8. explain the calculus of variations and apply it to the solution of problems;

Discipline Specific Skills and Knowledge:
9. apply analytical and numerical skills in mathematics;

Personal and Key Transferable / Employment Skills and Knowledge:
10. formulate and tackle problems in a logical and systematic manner;
11. present work clearly with justification of techniques and methods;
12. work co-operatively with peers and with the demonstrators to solve guided problems.

SYLLABUS PLAN - summary of the structure and academic content of the module
I. Probability theory
1. Random variables
2. Conditional probability
3. Probability distributions
• Discrete
• Continuous
II. Lagrangian formulation of classical mechanics
1. Calculus of variations
2. Euler-Lagrange equations
III. Solution of linear partial differential equations
1. Simple second order differential equations and common varieties: Harmonic oscillator, Schrödinger equation, Poisson's equation, wave equation and diffusion equation.
2. Separation of variables: The Laplacian family of equations in physics, separation of variables, mechanics of the technique, form of solutions, general solutions in series form, relation to Fourier series, spatial boundary conditions, time dependence, initial conditions.
3. Examples: rectangular drum, classical and quantum harmonic oscillator, waves at a boundary, temperature distributions, wavepacket/quantum particle in a box
4. Role of symmetry: Cylindrical and spherical polar co-ordinates, appearance of special functions. Use of special functions by analogy to sin, cos, sinh, cosh etc.
5. Examples: circular drum, hydrogen wave function
IV. Linear Algebra
1. Revision: Row and column vectors, matrices, matrix algebra, the solutions of systems of linear equations.
2. Eigenvalue equations I: The matrix equation Ax=ax, solving the matrix equation, the secular determinant, eigenvalues and eigenvectors, canonical form, normal modes/harmonics, simple coupled oscillators.
3. Eigenvalue equations II: Properties of eigenvectors: orthogonality, degeneracy, as basis vectors.
4. Eigenvalue equations III: Differential equations as eigenvalue equations and the matrix representation Ax=ax; choosing the basis, solving the equation, the secular determinant, eigenvalues and eigenvectors.
5. Examples: classical coupled modes, Schrödinger wave equation
6. Approximate solutions to differential equations (perturbation theory): use of eigenvectors, first- and second-order through repeated substitution, problem of degeneracies.
7. Examples: quantum particle in a well, a mass on drum, coupled particles
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 36 114
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning & teaching activities 22 hours 22×1-hour lectures Guided independent study 16 hours 8×2-hour self-study packages Guided independent study 30 hours 10×3-hour problems sets Scheduled learning & teaching activities 11 hours Problems class support Scheduled learning & teaching activities 3 hours Tutorial support Guided independent study 68 hours Reading, private study and revision

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Exercises set by tutor (0%) 6×30-minute sets (typical) (Scheduled by tutor) 1-10 Discussion in tutorials
Guided self-study (0%) 8×2-hour packages (Fortnightly) 1-10 Discussion in tutorials

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
10 × Problems sets 20% 3 hours per set (fortnightly) 1-10 Marked and discussed in problems class
Mid-term Test 1 15% 30 minutes (Term 1, Week 6) 1-9 Marked, then discussed in tutorials
Mid-term Test 2 15% 30 minutes (Term 2, Week 6) 1-9 Marked, then discussed in tutorials
Final Examination 50% 120 minutes (May/June assessment period) 1-9 Mark via MyExeter, collective feedback via ELE and solutions.

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
Whole module Written examination (100%) 1-9 August/September assessment period

Re-assessment is not available except when required by referral or deferral.

RE-ASSESSMENT NOTES
An original assessment that is based on both examination and coursework, tests, etc., is considered as a single element for the purpose of referral; i.e., the referred mark is based on the referred examination only, discounting all previous marks. In the event that the mark for a referred assessment is lower than that of the original assessment, the original higher mark will be retained.

Physics Modules with PHY Codes
Referred examinations will only be available in PHY3064, PHYM004 and those other modules for which the original assessment includes an examination component - this information is given in individual module descriptors.
RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener
ELE:

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Riley, K.F. Hobson, M.P., Bence, S.J. Mathematical Methods for Physics and Engineering 3rd Cambridge University Press 2006 978-0521679718
Extended Boas, M.L. Mathematical Methods in the Physical Sciences 3rd John Wiley and Sons 2005 978-0-471-36580-8
Extended Gregory, R.D. Classical Mechanics Cambridge University Press 2006 0-521-534097
Extended James, G. Advanced modern engineering mathematics Addison-Wesley 1993 0-201-56519-6
Extended Kreyszig, E. Advanced Engineering Mathematics 9th edition Wiley 2005 978-0-471-72897-9
Extended Spiegel, M.R. Advanced Mathematics for Engineers and Scientists (Schaum Outline Series) McGraw-Hill 1971 0-070-60216-6
Extended Stroud, K.A Engineering Mathematics 7th Palgrave Macmillan 2013 978-1-137-03120-4
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES PHY1026
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 5 No Thursday 15th December 2011 Thursday 26th January 2023
KEY WORDS SEARCH Physics; Equation; Differentiation; Eigenvalue; Solution; Eigenvectors; Matrix; Probabilities; Wave; Variables; Differential equations.

Please note that all modules are subject to change, please get in touch if you have any questions about this module.