basic training in mathematics a fitness program for science students r shankar 1995 edition

Basic Training in Mathematics A Fitness Program for Science Students Basic Training in Mathematics A Fitness Program for Science Students R.SHANKAR Yale University New Haven, Connecticut SPRINGER SCIENCE+BUSINESS MEDIA, LLC Library of Congress Cataloging-in-Publication Data On file ISBN 978-0-306-45036-5 DOI 10.1007/978-1-4899-6798-5 ISBN 978-1-4899-6798-5 (eBook) © Springer Science+Business Media New York 1995 Originally published by Plenum Press, New York in 1995 Softcover reprint of the hardcover 1st edition 1995 1098765432 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher For UMA PREFACE This book is based on a course I designed a few years ago and have been teaching at Yale ever since It is a required course for physics majors, and students wishing to skip it have to convince the Director of Undergraduate Studies of their familiarity with its contents Although it is naturally slanted toward physics, I can see a large part of it serving the needs of anyone in the physical sciences since, for the most part, only very basic physics ideas from Newtonian mechanics are employed The raison d'etre for this book and the course are identical and as follows While teaching many of the core undergraduate courses, I frequently had to digress to clear up some elementary mathematical topic which bothered some part of the class For instance, I recall the time I was trying to establish how ubiquitous the harmonic oscillator was by showing that the Taylor series of any potential energy function at a stationary point was given to leading order by a quadratic function of the coordinate At this point some students wanted to know what a Taylor series was A digression to discuss Taylor series followed At the next stage, when I tried to show that ifthe potential involved many coordinates, the quadratic approximation to it could be decoupled into independent oscillators by a change of coordinates, I was forced to use some form of matrix notation and elementary matrix ideas, and that bothered some other set of students Once again we digressed Now, I was not averse to the idea that in teaching physics, one would also have to teach some new mathematics For example, the course on electricity and magnetism is a wonderful context in which to learn about Legendre polynomials On the other hand, it is not the place to learn for the first time what a complex exponential like eim.P means Likewise, in teaching special relativity one does not want to introduce sinh and cosh, one wants to use them and to admire how naturally they serve our purpose To explain what these functions are at this point is like explaining a pun In other words, some of the mathematical digressions were simply not desirable and quite frustrating for the teacher and student alike Now, this problem was, of course, alleviated as the students progressed through the system, since they were taking first-rate courses in the mathematics department in the meantime and could soon tell you a surprising thing or two about the edgeof-the-wedge theorem But one wished the students would have a grasp of the basics of each essential topic at some rudimentary level from the outset, so that instructors could get on with their job with the least amount of digressions From the student's point of view, this allowed more time to think about the subject proper and more freedom to take advanced courses When this issue was raised before the faculty, my sentiments were shared by many It was therefore decided that I would design and teach a course that would deal with topics in differential calculus of one or more variables (including vii viii Preface trigonometric, hyperbolic, logarithmic, and exponential functions), integral calculus of one and many variables, power series, complex numbers and function of a complex variable, vector calculus, matrices, linear algebra, and finally the elements of differential equations In contrast to the mathematical methods course students usually take in the senior year, this one would deal with each topic in its simplest form For example, matrices would be two-by-two, unless a bigger one was absolutely necessary (say, to explain degeneracy) On the other hand, the treatment of this simple case would be thorough and not superficial The course would last one semester and be selfcontained It was meant for students usually in the sophomore year, though it has been taken by freshmen, upper-class students, and students from other departments This book is that course Each department has to decide if it wants to devote a course in the sophomore year to this topic My own view (based on our experience at Yale) is that such a preventive approach, which costs one course for just one semester, is worth hours of curing later on Hour for hour, I can think of no other course that will yield a higher payoff for the beginning undergraduate embarked on a career in the physical sciences, since mathematics is the chosen language of nature, which pervades all quantitative knowledge The difference between strength or weakness in mathematics will subsequently translate into the difference between success and failure in the sciences As is my practice, I directly address the student, anticipating the usual questions, imagining he or she is in front of me Thus the book is ideal for self-study For this reason, even a department that does not have, as yet, a course at this level, can direct students to this book before or during their sophomore year They can tum to it whenever they run into trouble with the mathematical methods employed in various courses Acknowledgments I am pleased to thank all the students who took Physics 30 l a for their input and Ilya Gruzberg, Sentil Todadri, and George Veronis for comments on the manuscript As always, it has been a pleasure to work with the publishing team at Plenum My special thanks to Senior Editor Amelia McNamara, Editor Ken Howell, and Senior Production Editor Joseph Hertzlinger I thank Meera and AJ Shankar for their help with the index But my greatest debt is to my wife Urns Over the years my children and I have been able to flourish, thanks to her nurturing efforts, rendered at great cost to herself This book is yet another example of what she has made possible through her tireless contributions as the family muse It is dedicated to her and will hopefully serve as one tangible record of her countless efforts R Shankar Yale University New Haven, Connecticut NOTE TO THE INSTRUCTOR If you should feel, as I myself do, that it is not possible to cover all the material in the book in one semester, here are some recommendations • To begin with, you can skip any topic in fine print I have tried to ensure that this does not violence to continuity The fine print is for students who need to be challenged or for a student who, long after the course, begins to wonder about some subtlety; or runs into some of this material in a later course, and returns to the book for clarification At that stage, the student will have the time and inclination to read the fine print • The only chapter which one can skip without any serious impact on the subsequent ones, is that on vector calculus It will be a pity if this route is taken; but it is better to leave out a topic entirely rather than rush through everything More moderate solutions like stopping after some sections, are also possible • Nothing teaches the student as much as problem solving I have given a lot of problems and wish I could have give more When I say more problems, I not mean more which are isomorphic to the ones given, except for a change of parameters, but genuinely new ones As for problems that are isomorphic, you can generate any number (say for a test) and have them checked by a program like Mathematica • While this course is for physical scientists, it is naturally slanted towards physics On the other hand, most of the physics ideas are from elementary Newtonian mechanics and must be familiar to anyone who has taken a calculus course You may still have to customize some of the examples to your specialty I welcome your feedback ix NOTE TO THE STUDENT In American parlance the expression "basic training" refers to the instruction given to recruits in the armed forces Its purpose is to ensure that the trainees emerge with the fitness that will be expected of them when they embark on their main mission In this sense the course provides basic training to one like yourself, wishing to embark on a program of study in the physical sciences It has been my experience that incoming students have a wide spectrum of preparation and most have areas that need to be strengthened If this is not done at the outset, it is found that the results are painful for the instructor and student alike Conversely, if you cover the basic material in this book you can look forward to a smooth entry into any course in the physical sciences Of course, you will learn more mathematics while pursuing your major and through courses tailored to your specialization, as well as in courses offered by the mathematics department This course is not a substitute for any of that But this course is unlike a boot camp in that you will not be asked to things without question; no instructor will bark at you to "hit that desk and give me fifty derivatives of ex." You are encouraged to question everything, and as far as possible everything you will be given a logical explanation and motivation The course will be like a boot camp in that you will be expected to work hard and struggle often, and will emerge proud of your mathematical fitness I have done my best to simplify this subject as much as possible (but no further), as will your instructor But finally it is up to you to wrestle with the ideas and struggle for total mastery of the subject Others cannot the struggling for you, any more than they can teach you to swim if you won't enter the water Here is the most important rule: as many problems as you can! Read the material before you start on the problems, instead of starting on the problems and jumping back to the text to pick up whatever you need to solve them This leads to patchy understanding and partial knowledge Start with the easy problems and work your way up This may seem to slow you down at first, but you will come out ahead Look at other books if you need to more problems One I particularly admire is Mathematical Methods in the Physical Sciences, by M Boas, published by Wiley and Sons, 1983 It is more advanced than this one, but is very clearly written and has lots of problems Be honest with yourself and confront your weaknesses before others do, as they invariably will Stay on top of the course from day one: in mathematics, more than anything else, your early weaknesses will return to haunt you later in the course Likewise, any weakness in mathematical preparation will trouble you xi xii Note to the Student during the rest of you career Conversely, the mental muscles you develop here will stand you in good stead 349 Differential Equations There is just one set of rules for dealing with them Assume a solution of the product fonn For example in the case of '1/J(x, y, z, t) assume '1/J(x, y, z, t) = X(x)Y(y)Z(z)T(t) Feed into the equation and decouple it into ordinary differential equations, one for each factor Solve each one subject to the boundary conditions Throw out options that blow up in the region of interest (at the origin, at infinity etc.) The most general solution is a linear superposition of such factorized solutions The coefficients in the linear superposition are detennined by boundary conditions ANSWERS Chapter 1.6.1 xj3- x /9,.0332224 (from series) and.0332222 (calculator) 1.6.2 (i) 3x cos(2 + :r ), (ii) -2 cos[cos(2:r )] sin(2:r ), (iii) sec :r tan2 :r, (iv) tanh:r, (v) 1/(1 + x ), (vi) 1/(1- x ), (vii) 0, (viii) 1/(1 1.6.3 100e· 12 ~ $112.72 1.6.4 v = 1.6.7 Square of side L/4 1.6.9 1,3 -! 1.6.12 1.6.13 x = is a minimum; :r = -1 is a maximum Chapter 2.1.2 :r lnx- x 2.2.1 sin- ~- sin- 2.2.3 2.2.5 ln(3/2) 2.2.6 1/3 2.2.7 1/(ln2) 2.2.9 /a= 1/(2a2 ), /4 = ~~ 2Gk G2 -k 2•2.) 1• (G~H~):I; (G2+k~)~ JSI + cosx) 3S2 Answers Chapter 3.1.2 /z = 3x + 2xy , / 11 = 5x y + 4y , fz 11 3.1.5 d = 4/ /5 3.2.5 I = M R /2, (3M R )/5 3.2.8 v = 12811" = 10xy4 = / 11z Chapter 4.2.3 Diverges 4.2.4 (i) ratio test is inconclusive, integral test says divergent (ii) r = e /27, convergent by ratio test (iii) Divergent by integral test (iv) Ratio test inconclusive, integral test says divergent (v) r = e, divergent (vi) Ratio test inconclusive, integral test says divergent 4.2.5 (i) C, (ii) C, (iii) C (iv) C 4.2.6 C, C, D, D, D 4.3.1 R = (i) /2, (ii) 1, (iii) 1, (iv) 1, (v) Converges for lxl > 1, (vi) lxl > 3v4 5v& ) 4.3.4.E=m (1 + 2v2 +s+16+ ,P=vE !· k: 4.3.6 T = 21rffg(1 + + ), 6T/T = 1/16 4.3.7 (i) ~[1 +x- x /2- x /6 +x /24], (ii) ~[1 +x +x /2+ x /6+ x /24], (iii) ln2 + x/2- x j8 +x j24- x j64 Chapter 137 761 5.2.3 z 35i7 - 35T7' 5.2.4 (i) Rez = 6/25, lmz = -8/25, lzl = 2/5, z• = 6/25 + 8/25i, 1/z = 3/2 + 2i, (ii) Rez = -7, Imz = 24, lzl = 25, z• = -7- 24i, 1/z = -(7/625}- (24j625)i, (iii) Rez = -7/25, lmz = 24/25, lzl = 1, z• = -(7/25) - (24/25)i, 1/ z = -(7/25) - (24/25) i, (iv) Rez = lmz = ~ jzj = 1/z = !.=f'i, 4, ¥ -i~, Answers 353 (v)Rez=cosO, lmz=sinO, 11z=z*=cosfl-isinfl 5.3.2 (i) Z1 = eiw/4, Z2 = 2e-iw/6, Z1Z2 = 2eiw/12, Z1IZ2 = 5e5wi/12 (ii) z = e2iarctan4/3, z2 = se-iw/2 5.3.3 lz1 + z2l 7.95, Phase is 98 radians 5.4.4 z = 1- i = _!2e-i"/4 , Io = 1001./2, current leads by 7rl4, resonance at w = 223.6 rads 5.4 Z 31t33i 41 = 5.4.6 z= l±iR( ,OR 1/( ,L)) · 5.4.7 Q(t) = e- 6t[4cos8t + 3sin8t]- 4cos10t Chapter 6.1.2 u., = -v11 u 11 = v., 6.1.3 f.,= /y 6.1.7 Poles at (±1 ± i)l./2, double pole at z = = ±i 6.1.8 Ur vslr, Vr = -uslr 6.1.12 (i) f = z 3, (ii) f = eiz, (iii) u is not harmonic 6.l.l3 Hint: Relate the Laplacian to 6.l.l4 Hint: Consider /2 a:::r• 6.2.3 sinxcoshy,cosxsinhy, Jsin2 x +sinh2 y, [x = n1r,y = 0] 6.2.4 z = n1r, (n + 112)7r, in1r, i(n + 112)7r 6.2.9 e2"in/N, n = 0, 1, N - Roots add to zero 6.2.10 + 4i, 12 + 5i 6.2.11 (i) ±(1 + i)l /2, (ii) ±(2 + i) 6.2.12 In + (2m + 1)i1r 6.2.13 (i) eosin + i sin In (ii) i1r 12 (iii) ±5ei"/4 (iv) ±eiw/3 6.2.14 (i) Repeated twice: ±i (ii) Repeated twice: [eiw/3, -1,e-iw/3] 6.2.15 J2ei""/ • The cube roots are (2) 116 (ei""/ 12 ,ei(w/12)±(2wi/3)) 6.2.16 e-wl2n+l/2) 6.4.1 1rle, -1re 6.4.5 (i) 1r I (2312), (ii) 0,0 (iii) 1r I 18, (iv)-1r /3, (v) 0, (vi) 1r I4 6.4.8 1r I (27e ) 6.5.3 Yes 354 Answers 6.5.4 No Chapter 7.4.3 (i) 516, 7.5.1 h = v-=R~2 -x"ll'2 -y72, Vh 7.5.4 Cylindrical: hp = 1, = -i \fR2-z2-fl2 : - j v Jn2-z2-fl2 hq, = p h = Spherical: hr = 1, rsinB h9 = r, hq, = 7.5.5 (ii) v'la, (iii) 12/ v'la, (iv) 7.5.7 (i) Towards the origin (ii) AT= -3v'2/10 7.5.8 (i) 2J3, (ii) v'3e , (iii) J3, (iv) All gradients were radial, this direction is perpendicular to radial 7.6.2 (i) no, (ii) yes, (iii) yes (iv) yes 7.6.3 (i) 1, (ii) 1, (iii) Possibly (iv) rjl = x y 7.6.4 (i) 1, (ii) rjl = x3y 7.6.5 (i) 1/2, (ii) no 7.6.11 (I) 0, (ii), 0, (iii) conservative, F = Vr/1, rjl = (x + y2)/2 7.6.13 211" 7.6.14 The curl has no component in the plane containing the contours for two line integrals 7.7.1 -sinx+2z 7.7.3 7.7.4 ~ 7.7.6 (i) 47rR • (ii) 21rR , (iii) Chapter 8.1 M +N ] = [ 10 12 M = 19 22 ] 50 IM' N) 8.3.4 {1, 2, -1); (-3, -4, 8) MN =[ 43 =[ [ 10 ] 15 22 -4 -12 ] 12 Answers 8.3.5 [ _1 ~~ ~~ ~2 ~2 l 3SS Chapter 9.I I yes, real scalars, no, U1 + U2 is not unitary, yes with integer field 9.1.2 yes, yes,no 9.1.5 No: 13} = 11}- 212) 9.2.1 (i) VJ = (2 + i(1- J3))j /2, V/1 = (J3 + 1)ij /2 (ii) VJ = (i -1)(i + J3)(v'6 -1)/4, Vl/ = + fii2 9.2.2 !(3i + 4j) ((104} + (78) )- 112 [104i- 78j) 9.2.4 When IV)= c!W} 9.2.6 Q = !jl {3 = - jfi ''Y = 9.5.2 [eigenvalue)(eigenvector components) (1) (0, 0, 1} (e±iB] ~(1, =fi, 0} 9.5.3 The answers are given in the same fonnat as the matrices in the assigned problem The three eigenvalues and eigenvectors for each matrix are given one below another *( (1) (1, 0, 0} [2] -5, -2, 1} [4) ?10(1, 0, 3} [-1) ~ (-1,0,1} (0] (0, 1,0} [1] ~(1,0,1) [-11-}a (1, -2, 1} [1]~ (-1,0,1) [2] 73(1, 1, 1} [0] ~(-1,0,1} [-v'2j !(1, - /2, 1) [ /2] }(1, /2, 1} [2] ~(-1,0,1) [2(1- /2)] ' !(1, - /2, 1) [2(1 + /2)] !(1, /2, 1} (2] ~(-J3,0,3) (3] (0, 1, 0) (6] !