# Prof Jey Sivaloganathan's teaching page

This page will contain information relating to courses I am teaching, copies of the tutorial sheets and summaries of the material covered to date.

## MA20218: Analysis 2A

This course will extend the notions of continuity and limit for real valued functions which you have come across in Analysis 1A to Rn, to normed spaces and to metric spaces, and develop some topology in these spaces. We will also study sequences of functions and their relation with integration and power series.

Course Diary

Week 1: Review of basic concepts of convergence of sequences and series

Handouts/Notes

Week 1

Problem Sheets

Sheet 0

Relevant text books include:

• "Calculus" by M. Spivak
• "Mathematical Analysis" by T.M. Apostol
• ...plus many more!

## Sobolev Spaces

This graduate course is run as part of the Taught Course Centre for the Mathematical Sciences organised in collaboration with Bristol, Imperial, Oxford and Warwick. (Follow this link for further information)

## MA30253: Continuum Mechanics

This course develops a general theory of continuum mechanics which can then be applied to modelling gases, fluids and elastic solids. In a continuum theory, we assume that the material is infinitely finely divisible, ignoring the atomic structure. This turns out to provide  realistic models and predictions at length scales much larger than the interatomic distance.  We then apply this theory to the study of inviscid fluids. (Unit description.)

Course Diary

Week 1: Introduction to continuum mechanics

Week 2 : Review of vector calculus, Einstein convention, kinematics of continua, material and spatial descriptions, homogeneous deformations

Week 3: Material derivative, streamlines and particle paths, local state of deformation, rate of change of deformation gradient, velocity gradient.

Week 4: Spin tensor, Rate of stretch tensor, Green strain tensor, vorticity

Week 5: Reynolds Transport Theorem, incompressible motions, continuity equation, circulation

Week 6: Transport Theorem (version 2), Transport Theorem for contour integrals, Balance of mass, Cauchy/Euler stress principle, Cauchy stress vector, Cauchy stress tensor, Balance of linear momentum

Week 7: Vortex lines and tubes, Kelvin's Circulation Theorem, Helmholtz Theorems on vorticity.

Week 8: Cauchy's Theorem on transport of vorticity, planar flows, velocity potential, stream function, complex velocity, complex potential,

Week 9: Examples of potential flows, Blasius Theorem, D'Alembert's paradox.

Week 10-11: Cartesian tensors and their properties.

Relevant text books include:

• "Mechanics of Continuous Media" by S.C. Hunter
• "A First Course in Continuum Mechanics" by O. Gonzalez and A.M. Stuart (online version of book available from the University of Bath library)
• "An Introduction to Continuum Mechanics" by M.E. Gurtin
• D.J. Acheson "Elementary fluid Dynamics"
• A. Chorin & J.E. Marsden "Mathematical introduction to fluid mechanics"
• D.E. Bourne & P.C. Kendall "Vector analysis & cartesian tensors"
• G.K. Batchelor "An introduction to fluid dynamics"

## MA40065: Viscous Fluid Mechanics

In this course we develop and study the fundamental principles underlying continuum mechanics and then apply these to the study of viscous fluids. (Unit description.)

Relevant text books include:

• "Mechanics of Continuous Media" by S.C. Hunter
• "A First Course in Continuum Mechanics" by O. Gonzalez and A.M. Stuart (online version of book available from the University of Bath library)
• "An Introduction to Continuum Mechanics" by M.E. Gurtin
• D.J. Acheson "Elementary fluid Dynamics"
• A. Chorin & J.E. Marsden "Mathematical introduction to fluid mechanics"
• D.E. Bourne & P.C. Kendall "Vector analysis & cartesian tensors"
• G.K. Batchelor "An introduction to fluid dynamics"

## MA40049: Elasticity

In this course, we develop the general principles of the theory of elasticity. The classical theory of linear elasticity can then be obtained by expanding the constitutive law relating stress to strain in a neighbourhood of the undeformed state.

Relevant text books include:

"Mechanics of Continuous Media" by S.C. Hunter

"Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity" by P.G. Ciarlet

"A First Course in Continuum Mechanics" by O. Gonzalez and A.M. Stuart (online version of book available from the University of Bath library)

"An Introduction to Continuum Mechanics" by M.E. Gurtin

"Nonlinear Elastic deformations" by R.W. Ogden

"Continuum Mechanics" by A.J.M. Spencer

Interesting practical aspects of materials science are contained in:

"The New Science of Strong Materials" by J.E. Gordon (first published 1968)

## Analytical and Geometric Theory of Differential Equations

This course has the intertwoven themes of ordinary differential equations, the Calculus of Variations, Lagrangian Systems and Hamiltonian Systems, and looks at some of the elegant and striking connections between them.

Hamilton’s Principle states that a system evolves in such a way that a given integral over the path, called the action integral, is stationary. Fermat’s Principle, that light rays travel through a material along paths that minimise the total travel time, is an example of Hamilton’s Principle. The condition that the given integral is minimised then leads to a differential equation, called the Euler-Lagrange equation(s), for the path itself. This approach is known generally as a  ‘variational principle’ and many familiar problems in physics, such as Newton’s laws of motion, can be recast in this way. The study of such variational problems constitutes the Calculus of Variations. In many cases, the Euler-Lagrange equations can in turn be recast as a first order autonomous system of differential equations, called a Hamiltonian System, which allows us to deduce general properties of solutions by studying the geometry of the flow induced by the differential equation(s) in the corresponding phase space.

The concepts and principles introduced in this course have led to many developments in modern pure mathematics (such as symplectic geometry and ergodic theory) and, besides applying to the equations of classical mechanics, they have motivated much of modern physics.

Exam Solutions for 2011-2012

Relevant text books include:

"Introduction to Analytical Dynamics" by N.M.J. Woodhouse (Online version of book available from University of Bath Library)

"Calculus of Variations"  by Gelfand and Fomin

"Mechanics"  by Landau and Lifshitz

"Optimization-Theory and Applications" by L. Cesari

"Variational Calculus and Optimal Control"  by J.L. Troutman

" Principles of Mechanics" by J.L. Synge and B.A. Griffiths

"Mathematical Methods of Classical Mechanics" by V.I. Arnold

"The Feynman Lectures on Physics",  Volume 2,  by R. Feynman

"Introduction to Dynamics"  by Percival and Richards

"Analytical Dynamics of Particles" by Whittaker

"Mechanics"  by V. Kibble

"A Geometrical Theory of ODEs" by V.I. Arnold

"A Treatise on Analytical Dynamics" by Pars