PHY2025 
Mathematics with Physical Applications 
201718 

Dr J. Bertolotti 


Delivery Weeks: 
T1:0111, T2:0111


Level: 
5 (NQF) 

Credits: 
15 NICATS / 7.5 ECTS 

Enrolment: 
146 students (approx) 

Description
This module will 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. The emphasis is on practical
skills rather than formal proofs. Students will acquire skills in some key 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.
Module Aims
This module predates the current template; refer to the description above and the following ILO sections.
Intended Learning Outcomes (ILOs)
A student who has passed this module should be able to:

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

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

Personal and Key Transferable / Employment Skills and Knowledge:
 formulate and tackle problems in a logical and systematic manner;
 present work clearly with justification of techniques and methods;
 work cooperatively with peers and with the demonstrators to solve guided problems.
Syllabus Plan

Probability theory
 Random variables
 Conditional probability
 Probability distributions
 Discrete
 Continuous

Lagrangian formulation of classical mechanics
 Calculus of variations
 EulerLagrange equations

Solution of linear partial differential equations
 Simple second order differential equations and common varieties: Harmonic oscillator, Schrödinger equation, Poisson's equation, wave equation and diffusion equation.
 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.
 Examples: rectangular drum, classical and quantum harmonic oscillator, waves at a boundary, temperature distributions, wavepacket/quantum particle in a box
 Role of symmetry: Cylindrical and spherical polar coordinates, appearance of special functions. Use of special functions by analogy to sin, cos, sinh, cosh etc.
 Examples: circular drum, hydrogen wave function

Linear Algebra
 Revision: Row and column vectors, matrices, matrix algebra, the solutions of systems of linear equations.
 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.
 Eigenvalue equations II: Properties of eigenvectors: orthogonality, degeneracy, as basis vectors.
 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.
 Examples: classical coupled modes, Schrödinger wave equation
 Approximate solutions to differential equations (perturbation theory): use of eigenvectors, first and secondorder through repeated substitution, problem of degeneracies.
 Examples: quantum particle in a well, a mass on drum, coupled particles
Learning and Teaching
Learning Activities and Teaching Methods
Description 
Study time 
KIS type 
22×1hour lectures 
22 hours

SLT 
8×2hour selfstudy packages 
16 hours

GIS 
10×3hour problems sets 
30 hours

GIS 
Problems class support 
11 hours

SLT 
Tutorial support 
3 hours

SLT 
Reading, private study and revision 
68 hours

GIS 
Assessment
Weight 
Form 
Size 
When 
ILOS assessed 
Feedback 
0% 
Exercises set by tutor 
6×30minute sets (typical) 
Scheduled by tutor 
110 
Discussion in tutorials

0% 
Guided selfstudy 
8×2hour packages 
Fortnightly 
110 
Discussion in tutorials

20% 
10 × Problems Sets 
3 hours per set 
Fortnightly 
110 
Marked and discussed in problems class

15% 
Midterm Test 1 
30 minutes 
Weeks T1:06 
19 
Marked, then discussed in tutorials

15% 
Midterm Test 2 
30 minutes 
Weeks T2:06 
19 
Marked, then discussed in tutorials

50% 
Final Examination 
120 minutes 
May/June assessment period 
19 
Mark via MyExeter, collective feedback via ELE and solutions. 
Resources
The following list is offered as an indication of the type & level of information that
students are expected to consult. Further guidance will be provided by the Module Instructor(s).
Core text:
Supplementary texts:

Boas M.L. (2005), Mathematical Methods in the Physical Sciences (3^{rd} edition), John Wiley and Sons, ISBN 9780471365808 (UL: 510 BOA)

Gregory R.D. (2006), Classical Mechanics, Cambridge University Press, ISBN 0521534097 (UL: 531 GRE)

James G. (1993), Advanced modern engineering mathematics, AddisonWesley, ISBN 0201565196 (UL: 510.2462 JAM)

Kreyszig E. (2005), Advanced Engineering Mathematics (9^{th} edition), Wiley, ISBN 9780471728979 (UL: 510.2462 KRE)

Spiegel M.R. (1971), Advanced Mathematics for Engineers and Scientists, Schaum Outline Series, McGrawHill, ISBN 0070602166 (UL: 510 SPI)

Stroud K.A. and Booth D.J. (2013), Engineering Mathematics (7^{th} edition), Paulgrave MacMillan, ISBN 9781137031204 (UL: 510.2462 STR)
ELE:
Further Information
Prior Knowledge Requirements
Prerequisite Modules 
Mathematics for Physicists (PHY1026) 
Corequisite Modules 
none 
Reassessment
Reassessment is not available except when required by referral or deferral.
Original form of assessment 
Form of reassessment 
ILOs reassessed 
Time scale for reassessment 
Whole module 
Written examination (100%) 
19 
August/September assessment period 
Notes: See Physics Assessment Conventions.
KIS Data Summary
Learning activities and teaching methods 
SLT  scheduled learning & teaching activities 
36 hrs 
GIS  guided independent study 
114 hrs 
PLS  placement/study abroad 
0 hrs 
Total 
150 hrs 


Summative assessment 
Coursework 
20% 
Written exams 
80% 
Practical exams 
0% 
Total 
100% 

Miscellaneous
IoP Accreditation Checklist 
 MT07 Solution of linear partial differential equations
 MT13 Probability distributions.

Availability 
unrestricted 
Distance learning 
NO 
Keywords 
Physics; Equation; Differentiation; Eigenvalue; Solution; Eigenvectors; Matrix; Probabilities; Wave; Variables; Differential equations. 
Created 
01Oct10 
Revised 
01Oct11 