Multivariable Calculus

by Strauss, Monty J.; Bradley, Gerald L.; Smith, Karl J.

• ISBN13: 9780130337856
• ISBN10: 0130337854
• Edition: 3rd
• Format: Paperback
• Copyright: 2002-01-01
• Publisher: Pearson College Div

Summary

Built from the ground up, to meet the needs of those learning calculus today, Bradley/Smith,Calculus was the first book to pair a complete calculus syllabus with the best elements of reform--like extensive verbalization and strong geometric visualization. The Third Edition of this groundbreaking book has been crafted and honed, making itthe book of choice for those seeking the best of both worlds. Numerous chapters offer an exciting choice of problem sets and include topics such as vectors in the plane and in space, vector-valued functions, partial differentiation, multiple integration, introduction to vector analysis, and introduction to differential equations. For individuals learning calculus for their futures in various engineering, science, or math fields.

Table of Contents

Preface	ix
Functions and Graphs	1	(48)
Preliminaries	2	(11)
Lines in the Plane	13	(6)
Functions and Graphs	19	(14)
Inverse Functions; Inverse Trigonometric Functions	33	(16)
Chapter 1 Review	42	(4)
Guest Essay: Calculus Was Inevitable, John Troutman	46	(2)
Mathematical Essays	48	(1)
Limits and Continuity	49	(48)
The Limit of a Function	50	(11)
Algebraic Computation of Limits	61	(9)
Continuity	70	(10)
Exponential and Logarithmic Functions	80	(17)
Chapter 2 Review	92	(5)
Differentiation	97	(86)
An Introduction to the Derivative: Tangents	98	(12)
Techniques of Differentiation	110	(9)
Derivatives of Trigonometric, Exponential, and Logarithmic Functions	119	(6)
Rates of Change: Modeling Rectilinear Motion	125	(13)
The Chain Rule	138	(8)
Implicit Differentiation	146	(11)
Related Rates and Applications	157	(8)
Linear Approximation and Differentials	165	(18)
Chapter 3 Review	177	(4)
Group Research Project: Chaos	181	(2)
Additional Applications of the Derivative	183	(88)
Extreme Values of a Continuous Function	184	(11)
The Mean Value Theorem	195	(6)
Using Derivatives to Sketch the Graph of a Function	201	(16)
Curve Sketching with Asymptotes: Limits Involving Infinity	217	(12)
l'Hopital's Rule	229	(9)
Optimization in the Physical Sciences and Engineering	238	(12)
Optimization in Business, Economics, and the Life Sciences	250	(21)
Chapter 4 Review	263	(6)
Group Research Project: Wine Barrel Capacity	269	(2)
Integration	271	(84)
Antidifferentiation	272	(10)
Area as the Limit of a Sum	282	(8)
Riemann Sums and the Definite Integral	290	(12)
The fundamental Theorems of Calculus	302	(7)
Integration by Substitution	309	(7)
Introduction to Differential Equations	316	(12)
The Mean Value Theorem for Integrals; Average Value	328	(6)
Numerical Integration: The Trapezoidal Rule and Simpson's Rule	334	(8)
An Alternative Approach: The logarithm as an Integral	342	(13)
Chapter 5 Review	346	(5)
Guest Essay: Kinematics of Jogging, Ralph Boas	351	(1)
Mathematical Essays	352	(1)
Cumulative Review: Chapters 1-5	353	(2)
Additional Applications of the Integral	355	(70)
Area between Two Curves	356	(6)
Volume	362	(13)
Polar Forms and Area	375	(10)
Arc length and Surface Area	385	(10)
Physical Applications: Work, Liquid Force, and Centroids	395	(12)
Applications to Business, Economics, and life Sciences	407	(18)
Chapter 6 Review	417	(7)
Group Research Project: Houdini's Escape	424	(1)
Methods of Integration	425	(68)
Review of Substitution and Integration by Table	426	(9)
Integration by Parts	435	(6)
Trigonometric Methods	441	(7)
Method of Partial Fractions	448	(9)
Summary of Integration Techniques	457	(4)
First-Order Differential Equations	461	(11)
Improper Integrals	472	(9)
Hyperbolic and Inverse Hyperbolic Functions	481	(12)
Chapter 7 Review	487	(4)
Group Research Project: Buoy Design	491	(2)
Infinite Series	493	(80)
Sequences and Their Limits	494	(11)
Introduction to Infinite Series: Geometric Series	505	(9)
The Integral Test, p-Series	514	(7)
Comparison Tests	521	(6)
The Ratio Test and the Root Test	527	(6)
Alternating Series; Absolute and Conditional Convergence	533	(11)
Power Series	544	(9)
Taylor and Maclaurin Series	553	(20)
Chapter 8 Review	566	(4)
Group Research Project: Elastic Tightrope	570	(1)
Cumulative Review: Chapters 6-8	571	(2)
Vectors in the Plane and in Space	573	(60)
Vectors in R2	574	(8)
Coordinates and Vectors in R3	582	(6)
The Dot Product	588	(9)
The Cross Product	597	(9)
Parametric Representation of Curves; lines in R3	606	(9)
Planes in R3	615	(7)
Quadric Surfaces	622	(11)
Chapter 9 Review	628	(4)
Group Research Project: Star Trek	632	(1)
Vector-Valued Functions	633	(60)
Introduction to Vector Functions	634	(8)
Differentiation and Integration of Vector Functions	642	(9)
Modeling Ballistics and Planetary Motion	651	(9)
Unit Tangent and Principal Unit Normal Vectors; Curvature	660	(13)
Tangential and Normal Components of Acceleration	673	(20)
Chapter 10 Review	679	(5)
Guest Essay: The Stimulation of Science, Howard Eves	684	(3)
Mathematical Essays	687	(1)
Cumulative Review: Chapters 1-10	688	(5)
Partial Differentiation	693	(84)
Functions of Several Variables	694	(7)
Limits and Continuity	701	(9)
Partial Derivatives	710	(10)
Tangent Planes, Approximations, and Differentiability	720	(9)
Chain Rules	729	(8)
Directional Derivatives and the Gradient	737	(12)
Extrema of Functions of Two Variables	749	(12)
Lagrange Multipliers	761	(16)
Chapter 11 Review	770	(5)
Group Research Project: Desertification	775	(2)
Multiple Integration	777	(82)
Double Integration over Rectangular Regions	778	(9)
Double Integration over Nonrectangular Regions	787	(8)
Double Integrals in Polar Coordinates	795	(9)
Surface Area	804	(8)
Triple Integrals	812	(10)
Mass, Moments, and Probability Density Functions	822	(11)
Cylindrical and Spherical Coordinates	833	(10)
Jacobians: Change of Variables	843	(16)
Chapter 12 Review	852	(5)
Group Research Project: Space-Capsule Design	857	(2)
Vector Analysis	859	(76)
Properties of a Vector Field: Divergence and Curl	860	(7)
Line Integrals	867	(10)
The Fundamental Theorem and Path Independence	877	(10)
Green's Theorem	887	(11)
Surface Integrals	898	(10)
Stokes' Theorem	908	(8)
The Divergence Theorem	916	(19)
Chapter 13 Review	924	(5)
Guest Essay: Continuous vs. Discrete Mathematics	929	(1)
Cumulative Review: Chapters 11-13	930	(1)
Mathematical Essays	931	(1)
Cumulative Review: Chapters 11-13	932	(3)
Introduction to Differential Equations	935	(1)
First-Order Differential Equations	936	(11)
Second-Order Homogeneous Linear Differential Equations	947	(10)
Second-Order Nonhomogeneous Linear Differential Equations	957	(9)
Chapter 14 Review	966	(3)
Group Research Project: Save the Perch Project	969
Appendices
A Introduction of The Theory of Limits	A-1
Selected Proofs	A-8
Significant Digits	A-17
Short Table of Integrals	A-21
Trigonometric Formulas	A-31
Answers to Selected Problems	A-34
Credits	A-54
Index	A-57

Excerpts

This text was developed to blend the best aspects of calculus reform with the reasonable goals and methodology of traditional calculus. It achieves this middle ground by providing sound development, stimulating problems, and well-developed pedagogy within a framework of a traditional structure. "Think, then do," is a fair summary of our approach. New to this Edition The acceptance and response from our first two editions has been most gratifying. For the third edition, we wanted to take a good book and make it even better. If you are familiar with the previous editions, the first thing you will notice is that we have added a new coauthor, Monty J. Strauss. His added expertise, and his attention to accuracy and detail, as well as his many years of experience teaching calculus, have added a new dimension to our exposition. Here is what is new in the third edition: Organization In this edition we introduce ex and In x in Chapter 2 after we have defined the notion of a limit. This is beneficial because it allows the number a to be properly defined using limits. We also assume a knowledge of the conic sections and their graphs. A freeStudent Mathematics Handbookis available that provides review and reference material on these transcendental functions. 1'Hopital's Rule is now covered earlier in Chapter 4. This placement allows instructors to explore more interesting applications like curve sketching. A new section covering applications to business, economics, and the life sciences has been added to Chapter 6 on Applications of the Integral. This new material is designed to help students see how calculus relates to and is used in other disciplines. The chapter on polar coordinates and parametric forms has been distributed to other chapters in the book. The polar coordinate system and graphing in polar forms is in Chapter 6 in the context of the integration topic of finding areas. Parametric representation of curves now appears in the book where it is first needed, in Chapter 9. Modeling continues as a major theme in this edition. Modeling is now introduced in Section 3.4, and then appears in almost every section of the book. These applications are designated MODELING PROBLEMS. Some authors use the words "Modeling Problem" to refer to any applied problem. In the third edition of Calculus, we make a distinction betweenmodeling problemsandapplication problemsby defining a modeling problem as follows. Amodeling problemis a problem that requires that the reader make some assumptions about the real world in order to derive or come up with the necessary mathematical formula or mathematical information to answer the question. These problems also include real-world examples of modeling by citing the source of the book or journal that shows the modeling process. Problem Sets We have added a new major category of problems, calledcounterexample problem.Acounterexampleis an example that disproves a proposition or theorem, and in mathematics we are often faced with a proposition that is true or false, and our task is to prove the proposition true or to find a counterexample to disprove the proposition. In the third edition ofCalculus,we attempt to build the student's ability with this type of situation to mean that the student must either find justification that the proposition is true or else find a counterexample. We believe this new form of problem to be important for preparing the student for future work in not only advanced mathematics courses, but also for analytically oriented courses. Exploration Problemsexplore concepts which may prove true or false and provide opportunities for innovative thinking. Interpretation Problemsrequire exposition that requires a line of thinking that is not directly covered in the textbook. Supplements Interactive