Study-Unit Description

Study-Unit Description


CODE PHY2211

 
TITLE Analytical Methods for Physicists

 
UM LEVEL 02 - Years 2, 3 in Modular Undergraduate Course

 
MQF LEVEL 5

 
ECTS CREDITS 6

 
DEPARTMENT Physics

 
DESCRIPTION Physical phenomena require appropriate modelling and mathematical techniques to identify key variables and build predictions. This study-unit continues to explore a diverse number of mathematical techniques which are essential in a Physicist's toolkit. While various techniques will be explored, each topic of discussion will be examined from a practical and physical viewpoint, mainly by exploring the physical interpretation of a particular quantity or technique, when and how the technique should be used, and identify its application in various topics in Physics. Thus, such tools will be explored through real-life contexts and applications to provide a more holistic understanding of the underlying physics.

The study-unit aims to cover the following topics and applications:

- An introduction to higher dimensional integrals in Cartesian and polar coordinate systems, and their applications (including but not limited to: averaging quantities, densities, area and volume, centre of mass);
- An introduction to vector transformations (with an emphasis on cylindrical and spherical polar coordinates), vector operators including the gradient, divergence, curl and Laplace together with their uses (with an emphasis on electromagnetism, gravitation and mechanics);
- An introduction to vector fields and potential theory including the notion of scalar and vector potentials;
- An introduction to line integrals and their application in the calculation of circulation;
- The notion of conservative and non-conservative vector fields and their application in calculating the work done along a trajectory;
- An introduction to surface integrals and their application in the calculation of flux;
- The statement and use of Green’s theorem and its relation to circulation and flux of a vector field, including applications to calculating work done and Ampere's circuitial law;
- The statement and use of the Divergence theorem and its relation with flux of a vector field and measure of a conserved quantity, including applications such as Gauss' law and Gaussian surfaces;
- The statement and use of Stokes’ theorem and its relation to circulation and flux of a vector field including applications such as Ampere's circuitial law;
- An introduction to Fourier transforms and their relation to waves and spectra, and their use in solving ordinary and partial differential equations, especially in the context of the Schrodinger equation, heat equation, wave equation, and forced or damped simple harmonic oscillators;
- An introduction to the use and properties of the Dirac Delta function and its applications (including, but not limited to, Coulomb's and Poisson's laws);
- A discussion on the Green's function, particularly in the context of Coulomb's and Poisson's laws.

Study-Unit Aims:

The study-unit aims to introduce students to various mathematical techniques that are essential in a physicists' arsenal. Furthermore, the study-unit further aims to link such techniques to relevant physical quantities and laws encountered in a diverse number of topics in Physics. This ultimately aims to provide students with the necessary tools to develop their fundamental understanding of complex physical concepts and hence be able to problem-solve various real-life scenarios.

Learning Outcomes:

1. Knowledge & Understanding:

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

- Use various higher dimensional integrals;
- Recognise the application of higher dimensional integrals in practical situations;
- State an expression for the gradient, divergence, curl and Laplace operators in Cartesian, polar, cylindrical and spherical coordinates;
- Define scalar and vector potentials;
- Use line and surface integrals;
- State the use of line and surface integrals in practical situations including work done, circulation and flux of a quantity;
- State when vector fields are conservative and their relationship with scalar potentials;
- Learn and identify the use of Green’s theorem in various physical situations;
- Learn and identify the use of the Divergence theorem in various physical situations especially in the context of electromagnetism and gravitation;
- Learn and identify the use of Stokes’ theorem in various physical situations especially in the context of electromagnetism and gravitation;
- Define the Fourier transform of a function;
- Recognise the use of the Fourier transform to solve various problems in Physics, including the Schrodinger equation, heat equation, wave equation, and forced or damped simple harmonic oscillators;
- State the Dirac Delta function and identify its applications in Physics;
- Learn about the Green's function and its use to solve differential equations.

2. Skills:

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

- Evaluate double and triple integrals including changing the order of integration and use of a Jacobian;
- Use double and triple integration to compute various physical quantities including mass, averages, area and volume, centre of mass;
- Evaluate the gradient, divergence, curl and Laplace operators in Cartesian, polar, cylindrical and spherical coordinates of scalar and vector fields;
- Determine when a vector field is conservative;
- Evaluate line and surface integrals, including their application in evaluating work done, circulation and flux of a quantity;
- Express a conservative field in terms of a scalar potential;
- Obtain the conservation of energy of a system using scalar and vector fields;
- State and apply Green’s theorem in various physical situations;
- State and apply the Divergence theorem in various physical situations especially in the context of electromagnetism and gravitation;
- State and apply Stokes’ theorem in various physical situations especially in the context of electromagnetism and gravitation;
- Define the Fourier and inverse Fourier transform of a function;
- Use the Fourier and inverse Fourier transform of various functions, particularly those commonly encountered in Physics such as the Heaviside step function, trigonometric functions, top-hat function;
- Use the Fourier transform to solve the Schrodinger equation, heat equation, wave equation, and forced or damped simple harmonic oscillators, and analyse their solutions;
- Define the properties of the Dirac Delta function;
- Use the Dirac Delta function including in the context of point charges or point masses;
- Define and about the Green's function of a differential equation and how it is used in the context of Physics.

Recommended Textbooks:

- Arfken, G. "Mathematical Methods for Physicists'', seventh Edition, Acamedic Press, New York (2003).
- 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: Knowledge of integration, differentiation and solving ordinary differential equations is required.

 
STUDY-UNIT TYPE Lecture and Tutorial

 
METHOD OF ASSESSMENT
Assessment Component/s Assessment Due Sept. Asst Session Weighting
Assignment SEM1 Yes 25%
Examination (3 Hours) SEM1 Yes 75%

 
LECTURER/S Gabriel Farrugia

 

 
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The availability of optional units may be subject to timetabling constraints.
Units not attracting a sufficient number of registrations may be withdrawn without notice.
It should be noted that all the information in the description above applies to study-units available during the academic year 2025/6. It may be subject to change in subsequent years.

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