Guide to Essential Math WHAT IS THE COMPLEMENTARY SCIENCE SERIES? The Complementary Science Series is an introductory, interdisplinary, and relatively inexpensive series of paperbacks for science enthusiasts. The series covers core subjects in chemistry, physics, and biological sciences but often from an interdisciplinary perspective. They are deliberately unburdened by excessive pedagogy, which is distracting to many readers, and avoid the often plodding treatment in many textbooks. These titles cover topics that are particularly appropriate for self-study although they are often used as complementary texts to supplement stan- dard discussion in textbooks. Many are available as examination copies to professors teaching appropriate courses. The series was conceived to fill the gaps in the literature between conventional textbooks and monographs by providing real science at an accessible level, with minimal prerequisites so that students at all stages can have expert insight into important and foundational aspects of current scientific thinking. Many of these titles have strong interdisciplinary appeal and all have a place on the bookshelves of literate laypersons. More information on titles appearing in the Complementary Science Series may be viewed at http://books.elsevier.com We hope you will enjoy this book. Guide to Essential Math A review for Physics, Chemistry and Engineering Students S. M. Blinder University of Michigan Ann Arbor USA AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier ACADEMIC PRESS Academic Press is an imprint of Elsevier 84 Theobald’s Road, London WC1X 8RR, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA First edition 2008 Copyright ⃝c 2008, Elsevier Inc. 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 electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: In honor of our son Matthew Bryant Blinder on the occasion of his 21st birthday This page intentionally left blank To the Reader Let me first tell you how the idea for this book came about. Many years ago, when I was a young assistant professor, I enthusiastically began my college teaching in a junior-level course that was supposed to cover quantum mechan- ics. I was eager to share my recently acquired insights into the intricacies of this very fundamental and profound subject, which seeks to explain the structure and behavior of matter and energy. About five minutes into my first lecture, a student raised his hand and asked, “Sir, what is that funny curly thing?” The object in question was ∂. Also, within the first week, I encountered the following very handy—but unfortunately wrong—algebraic reductions: a a + a, ln(x+y) ln x+ ln y, (ex)2 ex2. x + y x y Thus began my introduction to “Real Life” in college science courses! All of you here—in these intermediate-level physics, chemistry, or engi- neering course—are obviously bright and motivated. You got through your freshman and sophomore years with flying colors—or at least reasonable enough success to still be here. But maybe you had a little too much fun dur- ing your early college years—which is certainly an inalienable privilege of youth! Those math courses, in particular, were often a bit on the dull side. Oh, you got through OK, maybe even with As or Bs. But somehow their content vii viii To the Reader never became part of your innermost consciousness. Now you find yourself in a junior-, senior-, or graduate-level science course, with prerequisites of three or four terms of calculus. Your professor assumes you have really mastered the stuff. On top of everything, the nice xs, ys, and zs of your math courses have become ξs, ψs, ∇s, and other unfriendly looking beasts. This is where I have come to rescue you! You do not necessarily have to go back to your prerequisite math courses. You already have, on some subcon- scious level, all the mathematical skills you need. So here is your handy little Rescue Manual. You can read just the parts of this book you think you need. There are no homework assignments. Instead, we want to help you do the problems you already have in your science courses. You should, of course, work through and understand steps we have omitted in presenting impor- tant results. In many instances, it is easier to carry out a multistep derivation in your own way rather than to try and follow someone else’s sequence of manipulations. Sy M. Blinder Ann Arbor June 2007 Contents To the Reader vii 1 Mathematical Thinking 1 1.1 The NCAA March Madness Problem . . . . . . . . . . . . . . . . . . . . . 2 1.2 Gauss and the Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Torus Area and Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Einstein’s Velocity Addition Law . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 The Birthday Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Fibonacci Numbers and the Golden Ratio . . . . . . . . . . . . . . . . . 7 √ 1.8 π in the Gaussian Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.9 Function Equal to Its Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.10 Log of N Factorial for Large N . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.11 Potential and Kinetic Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.12 Riemann Zeta Function and Prime Numbers . . . . . . . . . . . . . . . 14 1.13 How to Solve It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.14 A Note on Mathematical Rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ix