THE FLUID DYNAMICS of JAMES CLERK MAXWELL 1. Introduction
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Mathematical Tripos 2020/2021 Guide to Courses in Part Ia
MATHEMATICAL TRIPOS 2021/2022 GUIDE TO COURSES IN PART IA This booklet provides an introduction for new students, giving an outline of the first year with informal and non-technical descriptions of the courses. It is intended to supplement the more formal descriptions contained in the booklet Schedules of Lecture Courses and Form of Examinations. These and other Faculty documents for students taking the Mathematical Tripos are available from http://www.maths.cam.ac.uk/undergrad/ Revised 6th September 2021 1 Introduction The Mathematical Tripos consists of Parts IA, IB and II, taken in successive years, with an optional fourth year, Part III, taken by students who do sufficiently well. Those who successfully complete three years are eligible to graduate with a BA honours degree, while those who go on to complete the additional fourth year graduate with both BA honours and MMath degrees. Unlike many degree courses, the Mathematical Tripos is not modular (at least in its first three years). It is tightly structured, with no choice in the first year, some choice in the second year, and a very wide choice in the third year. The examinations (especially in the second and third years) are cross-sectional, meaning that instead of each lecture course having a dedicated examination paper, each examination paper has questions on many lecture courses. The flexibility that this allows students is considered by the Faculty to be one of the great strengths of the Tripos. This booklet provides an introduction for new students, with an outline of Part IA and informal descrip- tions of the courses. -
James Clerk Maxwell
James Clerk Maxwell JAMES CLERK MAXWELL Perspectives on his Life and Work Edited by raymond flood mark mccartney and andrew whitaker 3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Oxford University Press 2014 The moral rights of the authors have been asserted First Edition published in 2014 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2013942195 ISBN 978–0–19–966437–5 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. -
Hydrodynamic Instabilities in Laser Produced Plasmas
the united nations educational, scientific abdus salam and cultural organization international centre for theoretical physics international atomic energy agency SMR 1521/17 AUTUMN COLLEGE ON PLASMA PHYSICS 13 October - 7 November 2003 Hydrodynamic Instabilities in Laser Produced Plasmas S. Atzeni Universita' "La Sapienza" and INFM Roma, Italy These are preliminary lecture notes, intended only for distribution to participants. strada costiera, I I - 34014 trieste italy - tel.+39 040 22401 I I fax+39 040 224163 - [email protected] -www.ictp.trieste.it UNIVERSITA' DEGLI STUDI DI ROMA LA SAPIENZA DIPARTIMENTO DI ENERGETICA Hydrodynamic Instabilities in Laser Produced Plasmas Stefano Atzeni Dipartimento di Energetica Universita di Roma "La Sapienza" and INFM Via A. Scarpa, 14-16, 00161 Roma, Italy E-mail: [email protected] Autumn College on Plasma Physics The Abdus Salam International Centre for Theoretical Physics Trieste, 13 October - 7 November 2003 The following material is Chapter 8 of the book "Physics of inertial confinement fusion" by S. Atzeni and J. Meyer-ter-Vehn Clarendon Press, Oxford (to appear in Spring 2004) -io Ok) 8 HYDRODYNAMIC STABILITY This chapter is devoted to hydrodynamic instabilities. ICF capsule implosions are inherently unstable. In particular, the Rayleigh-Taylor instability (RTI) tends to destroy the imploding shell (see Fig. 8.1). It should be clear that the goal of central hot spot ignition depends most critically on how to control the instability. Fig. 8.1 This is a major challenge facing ICF. For this reason, large efforts have been undertaken over the last two decades to study all aspects of RTI and related instabilities. -
Courses in Part Ib of the Mathematical Tripos
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics COURSES IN PART IB OF THE MATHEMATICAL TRIPOS This document contains a list of all the courses which are examinable in Part IB of the Mathematical Tripos together with an informal description of each course and suggestions for preliminary reading. A formal syllabus is given in the booklet Schedules for the Mathematical Tripos. which can be obtained from the Faculty Office, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA (telephone: 01223 337968; e-mail: [email protected]). All the documentation (including Schedules for the Mathematical Tripos) is available on the WWW (http://www.maths.cam.ac.uk/). June 2, 2004 READ ME Five changes have been implemented for 2004/05. They are as follows. There are two new courses, • { Complex Analysis, which will be lectured in the Lent Term and examined in Part IB, May/June 2005. This will run in parallel to Complex Methods, and it is expected that students with mainly pure mathematical interests will take Complex Analysis and students with mainly ap- plied mathematical interests will take Complex Methods. However, the courses are timetabled so that you could attend both courses. A total of four questions will be set in the examination on the two courses, two of which will be on material common to both courses. { The Geometry course has been replaced by a 16-lecture Geometry course given in the Lent term. Topological and Metric Spaces, which was lectured in the Easter term 2004, will be examined for • the first time in Part IB, May/June 2005. -
Convective to Absolute Instability Transition in a Horizontal Porous Channel with Open Upper Boundary
fluids Article Convective to Absolute Instability Transition in a Horizontal Porous Channel with Open Upper Boundary Antonio Barletta * and Michele Celli Department of Industrial Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-051-209-3287 Academic Editor: Mehrdad Massoudi Received: 28 April 2017; Accepted: 10 June 2017; Published: 14 June 2017 Abstract: A linear stability analysis of the parallel uniform flow in a horizontal channel with open upper boundary is carried out. The lower boundary is considered as an impermeable isothermal wall, while the open upper boundary is subject to a uniform heat flux and it is exposed to an external horizontal fluid stream driving the flow. An eigenvalue problem is obtained for the two-dimensional transverse modes of perturbation. The study of the analytical dispersion relation leads to the conditions for the onset of convective instability as well as to the determination of the parametric threshold for the transition to absolute instability. The results are generalised to the case of three-dimensional perturbations. Keywords: porous medium; Rayleigh number; absolute instability 1. Introduction Cellular convection patterns may develop in an underlying horizontal flow under appropriate thermal boundary conditions. The classic setup for the thermal instability of a horizontal fluid flow is heating from below, and the well-known type of instability is Rayleigh-Bénard [1]. Such a situation can happen in a channel fluid flow or in the filtration of a fluid within a porous channel. Both cases are widely documented in the literature [1,2]. -
UTO in Biological Physics
University of Cambridge Department of Applied Mathematics and Theoretical Physics University Lectureship in Biological Physics Applications are invited for a University Lectureship in Biological Physics to be held in the Department of Applied Mathematics and Theoretical Physics (DAMTP). Candidates should have an outstanding record of theoretical research in biological physics or related areas. The successful candidate will be expected to carry forward a programme of research in biological physics, involving the application of physical principles and mathematical techniques to the understanding of living systems and ideally, have a strong interaction with experimental research. We welcome applications from those working in areas such as biological fluid dynamics and biomechanics, as well as, for example, physical aspects of development, evolution and ecology. The post is part of an on-going cross-departmental effort in Cambridge to promote connections between the physical, mathematical and life sciences. The successful applicant will join a substantial and growing research effort within DAMTP in biological physics and mathematical biology. For more information about this research see: http://www.damtp.cam.ac.uk/research The position University Lecturer is one of four grades of academic post at the University of Cambridge, the others being Senior Lecturer, Reader and Professor. Rapid promotion through the grades is possible for candidates whose achievements merit this. A University Lectureship in Cambridge is roughly equivalent to an Associate Professorship in the US. The person appointed will be required to contribute to teaching, examining and administration and to research, including the supervision of graduate students and generation of research funding. So far as formal lecturing is concerned, the average load in DAMTP is some 40 hours per year, typically one undergraduate course and one graduate-level course. -
Microwave Heating of Fluid/Solid Layers : a Study of Hydrodynamic Stability
Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, fulfillment of the order would involve violation of copyright law. Please Note: The author retains the copyright while the New Jersey Institute of Technology reserves the right to distribute this thesis or dissertation Printing note: If you do not wish to print this page, then select “Pages from: first page # to: last page #” on the print dialog screen The Van Houten library has removed some of the personal information and all signatures from the approval page and biographical sketches of theses and dissertations in order to protect the identity of NJIT graduates and faculty. ASAC MICOWAE EAIG O UI/SOI AYES A SUY O YOYAMIC SAIIY A MEIG O OAGAIO y o Gicis In this work we study the effects of externally induced heating on the dynamics of fluid layers, and materials composed of two phases separated by a thermally driven moving front. One novel aspect of our study, is in the nature of the external source which is provided by the action of microwaves acting on dielectric materials. -
Mathematical Tripos: IA Vector Calculus Contents
Mathematical Tripos: IA Vector Calculus Contents 0 Introduction i 0.1 Schedule . i 0.2 Lectures . i 0.3 Printed Notes . ii 0.4 Examples Sheets . iii 0.5 Previous Comments, Comments on Comments and Style of Lectures. iii 0.6 Books ................................................... iv 0.7 Revision . iv 0.7.1 Vectors . iv 3 0.7.2 Cylindrical polar co-ordinates (ρ, φ, z) in R . ......................... vi 3 0.7.3 Spherical polar co-ordinates (r, θ, φ) in R ............................ vi 0.7.4 Determinants. vi 1 Partial Differentiation 1 1.1 Scalar Functions of One Variable . 1 1.1.1 Limits . 1 1.1.2 Continuity . 1 1.1.3 Differentiability . 1 1.1.4 Taylor’s Theorem . 2 1.2 Functions of Several Variables . 2 1.2.1 Limits . 3 1.2.2 Continuity . 3 1.3 Derivatives of Vector Functions of One Variable . 4 1.3.1 Example . 4 1.4 Partial Derivatives . 4 1.4.1 Definition . 5 1.4.2 Examples . 5 1.4.3 Directional Derivatives . 7 1.5 Differentiability . 7 1.5.1 Scalar Functions of Two Variables . 7 1.5.2 Functions of Several Variables . 8 1.5.3 Properties of Differentiable Functions . 9 1.6 The Chain Rule and Change of Variables . 9 m m 1.6.1 The Chain Rule for x : R → R and f : R → R ....................... 9 ` m m 1.6.2 The Chain Rule for x : R → R and f : R → R ...................... 11 ` m m n 1.6.3 The Chain Rule for x : R → R and f : R → R ..................... -
Hydrodynamic Stability and Turbulence in Fibre Suspension Flows Mathias
Hydrodynamic stability and turbulence in fibre suspension flows by Mathias Kvick May 2012 Technical Reports from Royal Institute of Technology KTH Mechanics SE-100 44 Stockholm, Sweden Akademisk avhandling som med tillst˚andav Kungliga Tekniska H¨ogskolan i Stockholm framl¨aggestill offentlig granskning f¨oravl¨aggandeav teknologie licentiatsexamen den 12 juni 2012 kl 14.00 i Seminarierrummet, Brinellv¨agen 32, Kungliga Tekniska H¨ogskolan, Stockholm. c Mathias Kvick 2012 Universitetsservice US{AB, Stockholm 2012 Mathias Kvick 2012, Hydrodynamic stability and turbulence in fibre suspension flows Wallenberg Wood Science Center & Linn´eFlow Centre KTH Mechanics SE{100 44 Stockholm, Sweden Abstract In this thesis fibres in turbulent flows are studied as well as the effect of fibres on hydrodynamic stability. The first part of this thesis deals with orientation and spatial distribution of fibres in a turbulent open channel flow. Experiments were performed for a wide range of flow conditions using fibres with three aspect ratios, rp = 7; 14; 28. The aspect ratio of the fibres were found to have a large impact on the fibre orientation distribution, where the longer fibres mainly aligned in the streamwise direction and the shorter fibres had an orientation close to the spanwise direction. When a small amount of polyethyleneoxide (PEO) was added to the flow, the orientation distributions for the medium length fibres were found to ap- proach a more isotropic state, while the shorter fibres were not affected. In most of the experiments performed, the fibres agglomerated into stream- wise streaks. A new method was develop in order to quantify the level of ag- glomeration and the streak width independent of fibre size, orientation and concentration as well as image size and streak width. -
Advanced Probability University of Cambridge, Part III of the Mathematical Tripos Michaelmas Term 2006
Advanced Probability University of Cambridge, Part III of the Mathematical Tripos Michaelmas Term 2006 Gr´egory Miermont1 1CNRS & Laboratoire de Math´ematique, Equipe Probabilit´es, Statistique et Mod´elisation, Bt. 425, Universit´eParis-Sud, 91405 Orsay, France Contents 1 Conditional expectation 5 1.1 The discrete case ................................ 5 1.2 Conditioning with respect to a σ-algebra ................... 6 1.2.1 The L2 case ............................... 7 1.2.2 General case ............................... 7 1.2.3 Non-negative case ............................ 8 1.3 Specific properties of conditional expectation ................. 10 1.4 Computing a conditional expectation ..................... 11 1.4.1 Conditional density functions ..................... 11 1.4.2 The Gaussian case ........................... 11 2 Discrete-time martingales 13 2.1 Basic notions .................................. 13 2.1.1 Stochastic processes, filtrations .................... 13 2.1.2 Martingales ............................... 14 2.1.3 Doob’s stopping times ......................... 14 2.2 Optional stopping ................................ 15 2.3 The convergence theorem ............................ 17 2.4 Lp convergence, p > 1 .............................. 18 2.4.1 A maximal inequality .......................... 18 2.5 L1 convergence ................................. 20 2.6 Optional stopping in the UI case ....................... 20 2.7 Backwards martingales ............................. 21 3 Examples of applications of discrete-time martingales -
Early Indian Mathematical Pilgrims to Cambridge
Early Indian Mathematical Pilgrims to Cambridge K. Razi Naqvi Department of Physics, Norwegian University of Science and Technology N-7491 Trondheim, Norway During the nineteenth and the early part of the next century, the University of Cambridge (UoC) was the Mecca of Mathematcis for students from the British Isles and other parts of the British Empire. UoC concentrated on undergraduate teaching; its examination system, particularly the Mathematical Sciences Tripos (hereafter abbreviated as MathTrip), served as a launch pad for re- munerative and influential careers in law, church, politics, even medicine, and various adminis- trative organs of the British Raj. Most examiners were not engaged in research, and few examinees dreamt of qualifying as a high wrangler, or of becoming a creative mathematician. The purpose of this article is to scrutinise the performance of the few Indian students who completed one or both parts of MathTrip, the career choices they made after returning to India, and their efforts, if any, towards the diffusion of modern mathematics in Indian schools and colleges. Almost all of the returnees became functionaries of the colossal British bureaucracy. Rejuvenation of Indian math- ematics was carried out mostly by other enthusiasts, among whom Muslims are conspicuous by their scarcity. “Whether good mathematicians, when women and a handful of men from other they die, go to Cambridge, I do not parts of the British Empire were two mi- know. But it is well known that a large nority groups, whose members performed [?] number of men go there when they equally well. I am concerned here essen- are young for the purpose of being con- tially with those who came from the In- verted into senior wranglers and Smith’s dian subcontinent, and even out of this prizemen.” I would have been happier to small group, I will only speak of those who quote this remark if its author, the eccen- passed MathTrip during the period 1898– tric Oliver Heaviside, had used small in- 1909. -
Courses in Part Ii of the Mathematical Tripos
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics COURSES IN PART II OF THE MATHEMATICAL TRIPOS This document contains a list of all the courses which are examinable in Part II of the Mathematical Tripos together with an informal description of each course. A formal syllabus is given in the booklet Schedules for the Mathematical Tripos which is obtainable from the Mathematics Faculty Office. In some cases, a suggestion for preliminary reading is given here; if not, you can get a more detailed view of the course from browsing the books listed in the Schedules. NB: The information given here is correct at the date given below. Sometimes late changes, such as swapping `clashing' courses, are unavoidable. June 4, 2004 1 Introduction You will find here an informal description of the structure of Part II of the Mathematical Tripos, list of all courses that are examinable in Part II of the Mathematical Tripos including the terms they are lectured in and likely clashes, together with non-technical summaries and suggestions for preliminary reading. 2 Structure of Part II The structure of Part II may be summarised as follows: • There are two types of lecture courses, labelled C and D. C-courses are all 24 lectures, D-courses may be 16 or 24 lectures. There are 10 C-courses and 25 D-courses. There is in addition a Computational Projects course. • C-courses are intended to be straightforward, whereas D courses are intended to be more challeng- ing. • There is no restriction on the number or type of courses you may present for examination.