Study-Unit Description

Study-Unit Description


TITLE Mathematics for Physicists 2

LEVEL 02 - Years 2, 3 in Modular Undergraduate Course



DESCRIPTION This study-unit discusses a number of important mathematical techniques that are used in a number of physics topics. These include multiple integrals, transformation of coordinates in multiple integrals, Green’s theorem, Gauss’ theorem, Stokes’ theorem and orthogonal curvilinear coordinates. The theoretical framework behind this mathematical techniques is discussed in full. However, the emphasis will be on their applications through the use of various examples. These examples will preferentially be chosen from physical applications creating a link between this topic and the other units where the mathematical techniques are used.

Study-unit Aims:

The objective of this study-unit is to teach the candidate a number of mathematical techniques that find application in a number of physics topics. This will be carried out by first outlining the theory behind the mathematical methods. Then the techniques will be applied on a number of different examples. The examples will preferentially be derived from physical applications to create a link between the theory and the practical application. This is meant to teach when and how to use these mathematical techniques. More specifically the study will provide:

- A discussion on cylindrical and spherical polar coordinates;
- An introduction to line integrals, double integrals, triple integrals and multiple integrals;
- An introduction to the use and properties of the Kronecker Delta and the Dirac Delta function;
- An introduction to Fourier transforms;
- A discussion on the use of the Jacobian for the transformation of coordinates in multiple integrals;
- An introduction to vector operators including the gradient, divergence, curl and Laplace operators together with their uses;
- The statement and use of Green’s theorem;
- The statement and use of Gauss’ (Divergence) theorem;
- The statement and use of Stokes’ theorem;
- An introduction potential theory including the notion of scalar and vector potentials;
- An introduction to the property and use of orthogonal curvilinear coordinates;
- A discussion on Green's function;
- A discussion on series and limits.

Learning Outcomes:

1. Knowledge & Understanding
By the end of the study-unit the student will be able to:

- state an expression for the gradient, divergence, curl and Laplace operators in Cartesian coordinates;
- state Green’s theorem;
- state Gauss’ (Divergence) theorem;
- state Stokes’ theorem;
- distinguish between scalar and vector potentials;
- distinguish between converging and non converging series.

2. Skills
By the end of the study-unit the student will be able to:

- evaluate line, surface and volume integrals as well as general double and triple integrals;
- determine the Fourier transform and the inverse Fourier transform;
- determine the Jacobian of a transformation;
- use the Jacobian to transform the integration variables of a double and a triple integrals;
- evaluate the gradient, divergence, curl and Laplace operators;
- use the Kronecker Delta to solve problems;
- use the Dirac Delta function and its properties to evaluate integrals;
- use Green’s theorem to solve problems;
- use Gauss’ theorem to solve problems;
- use Stokes’ theorem to solve problems;
- transform coordinates between different orthogonal curvilinear coordinates including Cartesian, cylindrical and spherical polar coordinates;
- solve problems involving scalar and vector potentials;
- determine the Green's function and use it to find solutions to differential equations.

Main Text/s and any supplementary readings:

Recommended textbooks:

- Arfken, G. "Mathematical Methods for Physicists'', seventh Edition, Acamedic Press, New York (2003).

Supplementary readings:

- Riley, K. F., Hobson, M. P. and Bence, S. J. "Mathematical Methods for Physics and Engineering: A Comprehensive Guide'', third edition, Cambridge University Press (2006).

ADDITIONAL NOTES Pre-Requisite qualifications: Follows from: An advanced level in Mathematics; PHY1125


Assessment Component/s Assessment Due Resit Availability Weighting
Assignment SEM1 Yes 10%
Examination (2 Hours) SEM1 Yes 90%

LECTURER/S Iman Farhat

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Units not attracting a sufficient number of registrations may be withdrawn without notice.
It should be noted that all the information in the study-unit description above applies to the academic year 2019/0, if study-unit is available during this academic year, and may be subject to change in subsequent years.