Department: PhysicsModule co-ordinator: Prof. Vlado LazarovCredit value: 20 creditsCredit level: IAcademic year of delivery: 2021-22

- See module specification for other years: 2022-23

## Pre-requisite modules

## Co-requisite modules

- None

## Prohibited combinations

Occurrence | Teaching period |
---|---|

A | Autumn Term 2021-22 to Spring Term 2021-22 |

This module covers some of the most important areas of mathematical physics. Vector calculus provides the mathematics required to understand and use scalar and vector fields, providing the tools essential to allow interpretation of electric, magnetic and gravitational fields in 2nd and 3rd year courses. By deriving expressions for curl, div, grad in different coordinate systems and relating these to physical examples the laws of Gauss, Ampère and Faraday are expressed mathematically in both their macrophysical and differential forms. Finally the basic properties of both real and complex matrices and tensors are studied, how to manipulate them and use them in various physical situations. These have application in special relativity, quantum mechanics, and other areas of physics.

Building on Mathematics I in stage 1, this module is strongly aligned with stage 2 physics modules and provides the mathematical skills required to fully address their physics content. The module describes the solution of ordinary and partial differential equations used in quantum mechanics, electromagnetism and thermodynamics, emphasising their application to physical situations such as wave motion, thermal conduction, and the solution of the time dependent Schrödinger equation. The Fourier series is used to provide the mathematics required to interpret the periodic functions generated. A development of these mathematical ideas allows non-periodic functions to be analysed, using the Fourier transform. This has applications throughout physics, in evaluating signals, and in diffraction theory. The module also introduces the Frobenius method which can be applied to solve 2nd order differential equations in which coefficients are also functions of the independent variable. This is applied to solve the Quantum Linear Harmonic Oscillator and an explanation of the origin of Hermite polynomials is given.

Throughout the module, all mathematics is placed firmly within a physics context and works closely with supported modules in order to provide the essential skills and capabilities required. This facilitates a far deeper appreciation of the associated physics and substantially augments a student’s ability to solve the more advanced and complex problems common in both higher programme stages and employment.

Classical Mechanics is a beautiful area of physics that describes physics ranging in scale from the motion of the galaxies through the heavens to the scattering of particle beams by stationary target nuclei. Here we introduce the Lagrangian approach to classical mechanics, which allows a system’s equations of motion to be defined in terms of its kinetic and potential energies. We consider rigid body dynamics building on material covered in stage 1. We also apply the Lagrangian approach to Kepler’s Laws, demonstrating their effectiveness and elegance. Finally we will study Hamilton’s Principle, linking to the Hamiltonian as studied in Quantum Mechanics.

*Subject content*

- Understand the distinction between scalar and vector fields and how they are represented.
- Calculate the gradient of a scalar field and understand its physical meaning.
- Find the rate of change of a scalar field in any direction.
- Define and understand the physical meaning of a conservative field, giving examples.
- Calculate the amount of work done in moving along any path in a vector field.
- Understand the concept of divergence of a vector field expressing it mathematically.
- Define and understand the physical meaning of a solenoidal field, giving examples.
- Evaluate the flux of a vector field through any surface.
- Apply the divergence theorem and evaluate both the integrals involved.
- Understand the concept of the curl of a vector field expressing it mathematically.
- Understand the concept of circulation of a vector field expressing it mathematically.
- Evaluate the circulation of a vector field round any closed loop.
- Apply Stokes's theorem and evaluate both the integrals involved.
- Apply the Laplace operator
^{2}to a function of the form f(x,y,z). - Derive and use expressions for curl, div, grad and
^{2}in different coordinate systems. - Express the laws of Gauss, Ampère and Faraday mathematically in both their macrophysical and differential forms.
- Define various types of matrix and be able to multiply, divide, add and subtract matrices.
- Solve linear simultaneous equations using matrix methods.
- Solve sets of homogeneous and non-homogeneous equations.
- Define the rank of a matrix and understand its significance.
- Understand the concept of ill-conditioning and be able to relate it to simple simultaneous equations.
- Be able to define linear dependence and independence and be able to test for these conditions.
- Set up matrix equations and apply matrix methods to solve problems in various branches of classical physics.
- Apply matrix methods to effect rotations and translations of coordinates/axes/objects.
- Understand the concepts and properties of eigenvalues and eigenvectors.
- Diagonalise a simple matrix and be able to extract the eigenvalues and eigenvectors.
- Normalise the eigenvectors.
- Define what is meant by orthogonal Hermitian, Unitary and Normal matrices and be able to demonstrate their properties.
- Appreciate that cartesian tensors are expressed in terms of components referred to rectangular cartesian coordinate systems.
- Understand the meaning of a transformation matrix.
- Be able to define what is meant by the rank or order of a tensor.
- Be able to calculate the product of tensors.
- Understand what is meant by a scalar invariant.
- Be able to apply the principles of contraction.
- Understand what is meant by a symmetric and an anti-symmetric tensor.
- Be able to apply the quotient rule to a tensor.
- Appreciate that tensors are of primary importance in connection with coordinate transforms.
- solve 2nd order homogeneous and inhomogeneous ordinary differential equations
- apply the method of separation of variables to derive the solution of second order partial differential equations
- describe the process through which the Fourier series can be applied to a physical system in order to solve a partial differential equation
- analyse the role boundary conditions play in a solution of a partial differential equation and how this is manifested in a physical context
- determine Fourier coefficients and generate the Fourier series for a variety of simple repeating waveforms
- recognise how the symmetry of a waveform relates to the structure of the related Fourier series and apply this knowledge to simply a solution
- define the distinction between Fourier series and Fourier transforms
- explain the origins of reciprocal broadening as an example of convolution and appreciate its implications in measuring instruments
- demonstrate the solution of a second order differential equation using the method of Frobenius
- Describe the Lagrange equations and apply them to dynamical systems.
- Describe the situations in which conservation laws are valuable and set up the description of problems in a way which allows them to be used to solve the problems.
- Derive and apply Kepler's Laws and apply them to a range of both terrestrial and astrophysical problems.
- Define Hamilton's Principle of least action and interpret it in terms of interference between nearby paths.
- Apply methods of rotational dynamics to examples such as wheel balancing.
- Define what is meant by the centre of percussion and calculate its location for various objects.
- Recall and manipulate Hamilton's equations including relating the Hamiltonian to total energy.
- Derive and apply criteria for rotational stability.
- Solve a range of dynamical problems, particularly involving rotation, using Lagrange's and/or Hamilton's equations in combination with the fundamental equations of kinematics and by applying where relevant, conservation of momentum and/or energy.

*Academic and graduate skills*

- analyse a complex problem to determine the most appropriate techniques to apply to arrive at a solution
- integrate mathematical processes into the solutions of a range of problems encountered in other modules, during the final year project, and during a career or postgraduate qualification
- develop capacity to conduct independent study through the use of formative assessment offered through the VLE

*Syllabus*

- Fields: scalar and vector fields, definitions and representations.
- Gradient: definition of gradient;.grad in Cartesian coordinates, vector differential operator , calculation of gradient, directional derivative; use of gradient in physics - electric field, heat conduction, gravitational field; measure of work - line integral; conservative field and scalar potential; examples of conservative fields, level surfaces, evaluation of line integrals.
- Divergence: vector area and measure of flux; element of vector area on a sphere and a cylinder, surface integral and its evaluation; volume integral; concept of divergence of a vector field; mathematical expression for divergence; calculation of divergence; divergence theorem, differential form of Gauss’s Law; physical dimensions in div and surface and volume integrals; magnetic and solenoidal fields.
- Combination of operators: Laplace operator, Laplace’s equation.
- Curl: concept of curl of a vector field, mathematical expression for curl; conservative field revisited, calculation of curl; rigid body rotation and comparison with water down a plug hole; Stokes’s theorem; differential form of Ampère’s Law; summary of integral theorems and field properties.
- Polar coordinates: grad div curl and
^{2}in spherical polars and cylindrical polars. - Applications: time as an extra variable - the continuity equation, examples in fluid flow, electricity and heat conduction, the heat conduction equation; the diffusion equation, Ampère’s Law revisited, differential form of Faraday’s law of electromagnetic induction.
- Matrices: basic operations, addition, multiplication, division. Determination of the inverse matrix via Gauss elimination and calculation of the cofactor matrix. Determinants of a 2x2 and 3x3 linear matrix, the rank of a matrix, linear dependence, and definitions of orthogonal and orthonormal matrix, normalisation.
- Matrix applications: linear simultaneous equations, matrix operators, translation, rotation, and reflection. Eigenvalues and eigenvectors of a matrix. Definitions of orthogonal Hermitian, unitary and normal matrices.
- Tensors: basic algebra of tensors; symmetry and anti-symmetry of tensors; the contraction of a tensor; the quotient rule applied to tensors; the basic transformation of tensor components
- revision of 2nd order homogeneous and inhomogeneous ordinary differential equations
- introduction to 2nd order partial differential equations, their solutions written in terms of a Fourier series; the one dimensional wave equation; the time dependent Schrodinger equation; the Laplace equation in two dimensions; the Diffusion equation
- Frobenius method applied to solution of 2nd order differential equations
- Fourier Series: origins and mathematical form of Fourier series, calculation of Fourier coefficients, example of square wave; use of symmetry; orthogonal functions; use of complex exponentials; applications to physics
- Fourier transforms: mathematical definition, relationship with Fourier series, inverse Fourier transform, Fourier transform pair, Fourier integrals; definition of odd and even functions. Time series analysis of a finite wave train, reciprocal broadening; convolution and the convolution theorem; Dirac delta-function; Fourier transform of a Gaussian; Fourier transforms applied to diffraction, Young’s slits
- Lagrange’s Equations including the following: Generalised coordinates and forces, Holonomic constraints, The Lagrangian and Lagrange’s equations, Canonical momenta, ignorable coordinates and Noether’s theorem
- Kepler’s Laws including the following: Equation of motion in polar coordinates, Conservation of the angular momentum in central field, Derivation of Kepler II: equal areas in equal times, Derivation of Kepler I: orbits are elliptical or hyperbolic, Derivation of Kepler III: period-squared proportional to major-axis-cubed, Applicability to electrostatic force fields including Rutherford scattering, Classical picture of the atom and gravity waves
- Rigid body dynamics including: Mathematical description of fixed rotation about an axis, Moment of inertia tensor: principal axes and principal moments, Angular momentum and kinetic energy in terms of inertia tensor, Application: wheel balancing, Application: Centre of percussion of cricket bat, squash racket etc.
- Hamilton’s Principle and the Hamiltonian including: Principle of least action (Hamilton’s Principle), The Hamiltonian and Hamilton’s equations, Energy-Hamiltonian relationship, Dynamics of a spinning top (aka why do bicycles stay upright).

Task | Length | % of module mark |
---|---|---|

Essay/courseworkMaths II Assignment 1 |
N/A | 10 |

Essay/courseworkMaths II Assignment 2 |
N/A | 10 |

Online Exam - 24 hrs (Centrally scheduled)Maths II - Spring Exam |
8 hours | 40 |

Online Exam - 24 hrs (Centrally scheduled)Maths II - Summer Exam |
8 hours | 40 |

None

Task | Length | % of module mark |
---|---|---|

Essay/courseworkMaths II Assignment 1 |
N/A | 10 |

Essay/courseworkMaths II Assignment 2 |
N/A | 10 |

Online Exam - 24 hrs (Centrally scheduled)Maths II - Spring Exam |
8 hours | 40 |

Online Exam - 24 hrs (Centrally scheduled)Maths II - Summer Exam |
8 hours | 40 |

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

Stroud K A; Further engineering mathematics (MacMillan)***

Boas M L; Mathematical methods in the physical sciences, 3rd Ed 2002 (Wiley)***

Kreyszig E; Advanced engineering methods, 8th Ed (John Wiley)**

D.W.Jordan and P.Smith – Mathematical Techniques: An Introduction for the Engineering, Physical and Mathematical Sciences 4th edition (Oxford)**

D.E. Bourne and P.C. Kendall. Vector analysis and cartesian tensors 2nd edition or later. 1977 (Reinhold, New York)**

Jordan D.W. and Smith P.– Mathematical Techniques: An Introduction for the Engineering, Physical and Mathematical Sciences 4th edition (Oxford)**

L.D. Landau, E.M. Lifshitz, “Mechanics: Volume 1 (Course of Theoretical Physics)”, Butterworth-Heinemann; 3rd ed (1982)*