Advanced Calculus Demystified Demystified Senes Accounting Demystified Advanced Statistics Demystified AIgebra Demystified Alternative Energy Demystified Anatomy Demy si ified AS P.NET 2.0 Demystified Aslronomy Demystified Audio Demystified Biology Demystified Biotechnology Demystified Business Calculus Demystified Business Math Demystified Business Si at is tics Demystified C++ Demystified Calculus Demystified Chemistry Demystified College Algebra Demystified Corporate Finance Demystified Data Structures Demystified Databases Den tyst ified Differential Equations Demystified Digital Electronics Demystified Earth Science /demystified Electricity Demystified Electronics Demystified Environmental Science Demystified Everyday Math Demystified Eorensics Demystified Genetics Demystified Geometry Demystified Home Networking Demystified investing Demystified Java Demystified JavaScript Demystified Linear Algebra Demystified Macroeconomics Demystified Management Accounting Demystified Math Proofs Demystified Math Word Problems Demystified Medical Hilling and Coding Demystified Medical Terminology Demystified Meteorology Demystified Microbiology /demystified Microeconomics Demystified Nanotechnology Demystified Nurse Management Demystified OOP Demystified Options Demystified Organic Chemistry Demystified Personal Computing Demystified Pharmacology Demystified Physics Demystified Physiology Demystified Pre-A Igehra Demystified Precalcidus Demystified Probability Demystified Project Management Demystified Psy ch o !og\' Det nyst ifie d Quality Management Demystified Quantum Mechanics Demystified Relativity Demystified Robotics Demystified Signals and Systems Demystified Six Sigma Demystified SQL Demystified Statics and Dynamics Demystified Statistics Demystified Technical Math Demystified Trigonometry Demystified UML Demystified Visual Basic 2005 Demystified Visual C# 2005 Demystified XML Demystified Advanced Calculus Demystified David Bachman New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2007 by The McGraw-Hill Companies All rights reserved Manufactured in the United States of America Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher 0-07-151109-1 The material in this eBook also appears in the print version of this title: 0-07-148121-4 All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark Where such designations appear in this book, they have been printed with initial caps McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs For more information, please contact George Hoare, Special Sales, at george_hoare@mcgraw-hill.com or (212) 904-4069 TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc (“McGraw-Hill”) and its licensors reserve all rights in and to the work Use of this work is subject to these terms Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent You may use the work for your own noncommercial and personal use; 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If you’d like more information about this book, its author, or related books and websites, please click here To Stacy ABOUT THE AUTHOR David Bachman, Ph.D is an Assistant Professor of Mathematics at Pitzer College, in Claremont, California His Ph.D is from the University of Texas at Austin, and he has taught at Portland State University, The University of Illinois at Chicago, as well as California Polytechnic State University at San Luis Obispo Dr Bachman has authored one other textbook, as well as 11 research papers in low-dimensional topology that have appeared in top peer-reviewed journals Copyright © 2007 by The McGraw-Hill Companies Click here for terms of use For more information about this title, click here CONTENTS Preface Acknowledgments xi xiii CHAPTER Functions of Multiple Variables 1.1 Functions 1.2 Three Dimensions 1.3 Introduction to Graphing 1.4 Graphing Level Curves 1.5 Putting It All Together 1.6 Functions of Three Variables 1.7 Parameterized Curves Quiz 1 11 12 15 CHAPTER Fundamentals of Advanced Calculus 2.1 Limits of Functions of Multiple Variables 2.2 Continuity Quiz 17 17 21 22 CHAPTER Derivatives 3.1 Partial Derivatives 3.2 Composition and the Chain Rule 3.3 Second Partials Quiz 23 23 26 31 32 viii Advanced Calculus Demystified CHAPTER Integration 4.1 Integrals over Rectangular Domains 4.2 Integrals over Nonrectangular Domains 4.3 Computing Volume with Triple Integrals Quiz 33 33 38 44 47 CHAPTER Cylindrical and Spherical Coordinates 5.1 Cylindrical Coordinates 5.2 Graphing Cylindrical Equations 5.3 Spherical Coordinates 5.4 Graphing Spherical Equations Quiz 49 49 51 53 55 58 CHAPTER Parameterizations 6.1 Parameterized Surfaces 6.2 The Importance of the Domain 6.3 This Stuff Can Be Hard! 6.4 Parameterized Areas and Volumes Quiz 59 59 62 63 65 68 CHAPTER Vectors and Gradients 7.1 Introduction to Vectors 7.2 Dot Products 7.3 Gradient Vectors and Directional Derivatives 7.4 Maxima, Minima, and Saddles 7.5 Application: Optimization Problems 7.6 LaGrange Multipliers 7.7 Determinants 7.8 The Cross Product Quiz 69 69 72 75 78 83 84 88 91 94 CHAPTER Calculus with Parameterizations 8.1 Differentiating Parameterizations 8.2 Arc Length 95 95 100 ... Advanced calculus is an exciting subject that opens up a world of mathematics It is the gateway to linear algebra and differential equations, as well as more advanced mathematical subjects like analysis,... usually covered as part of a standard calculus sequence, coming just after the first full year Names of college classes that cover this material vary greatly Possibilities include advanced calculus, ... multivariable calculus, and vector calculus At schools with semesters the class may be called Calculus III At quarter schools it may be Calculus IV The best way to use this book is to read the material