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# Calculus Early Transcendentals MultiVariable

## Third Edition| ©2015 Jon Rogawski

**This alternative version of Rogawski and Adams’**The most successful calculus book of its gene...

*Calculus: Early Transcendentals*includes chapters 11-17 of the Third Edition, and is ideal for instructors who just want coverage of topics in multivariable calculus.**This alternative version of Rogawski and Adams’**The most successful calculus book of its generation, Jon Rogawski’s

*Calculus: Early Transcendentals*includes chapters 11-17 of the Third Edition, and is ideal for instructors who just want coverage of topics in multivariable calculus.*Calculus*offers an ideal balance of formal precision and dedicated conceptual focus, helping students build strong computational skills while continually reinforcing the relevance of calculus to their future studies and their lives. Guided by new author Colin Adams, the new edition stays true to the late Jon Rogawski’s refreshing and highly effective approach, while drawing on extensive instructor and student feedback, and Adams’ three decades as a calculus teacher and author of math books for general audiences.

The Third Edition is also a fully integrated text/media package, with its own dedicated version of WebAssign *Premium *that boasts a robust collection of interactive learning aids.

**Maximize Teaching and Learning with WebAssign Premium**

Macmillan Learning and WebAssign have partnered to deliver WebAssign *Premium* – a comprehensive and flexible suite of resources for your calculus course. Combining the most widely used online homework platform with authoritative textbook content and Macmillan’s esteemed Calctools, WebAssign *Premium* extends and enhances the classroom experience for instructors and students. Preview course content and sample assignments at www.webassign.net/whfreeman.

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*Calculus: Early Transcendentals* includes chapters 11-17 of the Third Edition, and is ideal for instructors who just want coverage of topics in multivariable calculus.

*Calculus*offers an ideal balance of formal precision and dedicated conceptual focus, helping students build strong computational skills while continually reinforcing the relevance of calculus to their future studies and their lives. Guided by new author Colin Adams, the new edition stays true to the late Jon Rogawski’s refreshing and highly effective approach, while drawing on extensive instructor and student feedback, and Adams’ three decades as a calculus teacher and author of math books for general audiences.

The Third Edition is also a fully integrated text/media package, with its own dedicated version of WebAssign *Premium *that boasts a robust collection of interactive learning aids.

**Maximize Teaching and Learning with WebAssign Premium**

Macmillan Learning and WebAssign have partnered to deliver WebAssign *Premium* – a comprehensive and flexible suite of resources for your calculus course. Combining the most widely used online homework platform with authoritative textbook content and Macmillan’s esteemed Calctools, WebAssign *Premium* extends and enhances the classroom experience for instructors and students. Preview course content and sample assignments at www.webassign.net/whfreeman.

Features

**Conceptual Insights**encourage the student to develop a conceptual understanding of calculus by explaining important ideas clearly but informally.

**Graphical Insights**enhance the students' visual understanding by making the crucial connection between graphical properties and the underlying concept.

**Reminders**in the margins link back to important concepts discussed earlier in the text.

Caution notes warn students of common pitfalls they can encounter in understanding the material.

**Historical Perspectives**are brief vignettes that place key conceptual discoveries and advancements in their historical settings. They give students a glimpse into past accomplishments of great mathematicians and an appreciation for their significance.

**Assumptions Matter**uses short explanations and well-chosen counterexamples to help students appreciate why hypotheses are needed in theorems.

**Section Summaries**summarize a section’s key points in a concise and useful way to emphasize for students what is most important in the section.

**Section Exercise Sets**offer a comprehensive set of exercises closely coordinated with the text. These exercises vary in levels of difficulty from routine, to moderate, to more challenging. Also included are questions appropriate for written response or use of technology:

•

**Preliminary Exercises**begin each exercise set and need little or no computation. They can be used to check understanding of key concepts of a section before problems from the exercise set are assigned.

•

**Exercises**offers numerous problems from the routine drill problems to moderately challenging problems. These are carefully graded and include many innovative and interesting geometric and real world applications.

•

**Further Insights and Challenges**are more challenging problems that help to extend a section’s material.

•

**End of Chapter Review Exercises**offer a comprehensive set of exercises closely coordinated with the chapter material to provide additional problems for self study or assignments.

New to This Edition

**New author, Colin Adams**

Colin Adams is an award-winning teacher, widely read author, and distinguished researcher. A user of Jon Rogawski’s textbook, he brings his own classroom experience to the project, as well as a well-regarded ability to make calculus more engaging and meaningful to students without sacrificing its precision and rigor.

**The exercise sets were reviewed extensively by longtime users to ensure the utmost accuracy, clarity, and complete content coverage. Exercise sets were also modified to improve upon the grading by level of difficulty and to ensure even/odd pairing.**

Refined Exercises

Refined Exercises

In addition, numerous new exercises have been added throughout the text, particularly where new applications are available or to enhance conceptual development.

**, including**

New Examples

New Examples

Example 7 in Section 4.3 where the local minimum occurs at critical point but derivative does not exist

Example 5 in Section 6.1 that computes the areas between two curves in two ways

Several new examples in Chapter 10 including Example 1 (Fibonacci sequence) and Example 4 (bounded sequence) in Section 10.1 and Example 5 (repeated decimal expansion) in Section 10.2

Several examples in Chapter 14 including Example 7 (a limit that does not exist because different paths product different limits) in Section 14.2, Example 9 (gradient vectors perpendicular to level curves) in Section 14.5 and Example 5 (Second Derivative Tests fails) in Section 14.7

New Content Based on User and Reviewer Feedback

New Content Based on User and Reviewer Feedback

Coverage of these concepts now focuses more on concepts and methods, rather than formulas and memorization:

Quadric surfaces (Ch. 12)

**This edition includes a number of new figures that help students visualize concepts, including illustrations that explain:**

New Illustrations

New Illustrations

What a limit means (Fig. 5 in Section 2.1, Fig. 3 in Section 2.2)

The Intermediate Value Theorem (Fig. 2 in Section 2.8)

Vertically and horizontally simple regions (Figs. 2 and 9 in Section 6.1)

**Standardized Notation**

Notational changes bring this edition in line with standard notation usage in mathematics and other fields that use mathematics, presenting a consistent message to students. Other notational changes make it easier for students to comprehend the concepts.

For example, in multivariable chapters, notation for vector-valued functions is now written r(t) = <x(t), y(t)> instead of c(t) = (x(t), y(t)) and the standard notation V is used for potential functions.

**Developed with extensive feedback from instructors and students, W. H. Freeman’s breakthrough online course space offers:**

LaunchPad

LaunchPad

**Pre-built Units for each chapter**, curated by experienced educators, with media for that chapter organized and ready to assign or customize to suit your course.**All online resources for the text in one location**, including an interactive e-book, LearningCurve adaptive quizzing (see below), interactive applets, Dynamic Figures with manipulable variables, CalcClips whiteboard videos, and more.**Powerful Online Homework Options**, with algorithmically generated exercises including, precalculus quizzes, and more**Helpful analytics**, with a Gradebook that lets you see how your class is doing individually and as a whole.**A streamlined and intuitive interface**that lets you build an entire course in minutes.

LearningCurve

In a game-like format, LearningCurve adaptive and formative quizzing provides an effective way to get students involved in the coursework. It offers:

**A unique learning path for each student**, with quizzes shaped by each individual’s correct and incorrect answers.**A Personalized Study Plan,**to guide students’ preparation for class and for exams.**Feedback for each question**with live links to relevant e-book pages, guiding students to the reading they need to do to improve their areas of weakness.

**In addition to the robust online homework system in LaunchPad, instructors can take advantage of the following W. H. Freeman partnerships:**

ONLINE HOMEWORK OPTION

ONLINE HOMEWORK OPTION

**WeBWorK**

**webwork.maa.org**

W. H. Freeman offers approximately 2,500 algorithmically generated questions (with full solutions) through this free open source online homework system developed at the University of Rochester. Adopters also have access to a shared national library test bank with thousands of additional questions, including 1,500 problem sets correlated to the Third Edition.

**WebAssign Premium**

www.webassign.net/whfreemanWebAssign

www.webassign.net/whfreeman

Premium for Calculus, Third Edition integrates the book’s exercises into the world’s most popular and trusted online homework system, making it easy to assign algorithmically generated homework and quizzes. WebAssign Premium also offers access to all of the book’s digital resources, with the option of including the complete e-Book.

"The clarity of examples, as well as their interconnectedness, remains a strong point. This fact alone goes a long way toward helping students better learn the concepts."

--Erik Tou, instructor,Carthage College

“It strikes the right balance between readability for the student and rigor for the instructor.”

--Debra Carney, instructor,Colorado School of Mines"It is refreshing to see that even in the chapter on limits, there are application problems from a variety of disciplines, including engineering, physics and biology. The applications feel realistic and relevant as opposed to constructed and stilted."

--Maria Siopsis, instructor,Maryville College"The strenghts are in the Conceptual and Graphical Insights. These are the kinds of comments that can make things 'click' and fall into place for the students."

--Berit Givens, instructor,California State Polytechnic University, Pomona"The notation is a strong point of this book. It is used consistently and the authors do not shy away from using math instead of excess prose. Those two features alone put these chapters well ahead of my present text.""It's an invitation to learn calculus the right way."

--Jonathan Pearsall, instructor,College of Southern Nevada

--Nadjib Bouzar, instructor,University of Indianapolis

**
Calculus Early Transcendentals MultiVariable**

Third Edition| ©2015

Jon Rogawski

# Digital Options

**Calculus Early Transcendentals MultiVariable**

Third Edition| 2015

Jon Rogawski

## Table of Contents

**Rogawski/Adams: Calculus Early Transcendentals 3e, Multivariable Table of Contents**

Chapter 11: Parametric Equations, Polar Coordinates, and Conic Sections

11.1 Parametric Equations

Chapter 11: Parametric Equations, Polar Coordinates, and Conic Sections

11.2 Arc Length and Speed

11.3 Polar Coordinates

11.4 Area and Arc Length in Polar Coordinates

11.5 Conic Sections

Chapter Review Exercises

**Chapter 12: Vector Geometry**

12.1 Vectors in the Plane

12.2 Vectors in Three Dimensions

12.3 Dot Product and the Angle Between Two Vectors

12.4 The Cross Product

12.5 Planes in Three-Space

12.6 A Survey of Quadric Surfaces

12.7 Cylindrical and Spherical Coordinates

Chapter Review Exercises

**Chapter 13: Calculus of Vector-Valued Functions**

13.1 Vector-Valued Functions

13.2 Calculus of Vector-Valued Functions

13.3 Arc Length and Speed

13.4 Curvature

13.5 Motion in Three-Space

13.6 Planetary Motion According to Kepler and Newton

Chapter Review Exercises

**Chapter 14: Differentiation in Several Variables**

14.1 Functions of Two or More Variables

14.2 Limits and Continuity in Several Variables

14.3 Partial Derivatives

14.4 Differentiability and Tangent Planes

14.5 The Gradient and Directional Derivatives

14.6 The Chain Rule

14.7 Optimization in Several Variables

14.8 Lagrange Multipliers: Optimizing with a Constraint

Chapter Review Exercises

**Chapter 15: Multiple Integration**

15.1 Integration in Two Variables

15.2 Double Integrals over More General Regions

15.3 Triple Integrals

15.4 Integration in Polar, Cylindrical, and Spherical Coordinates

15.5 Applications of Multiple Integrals

15.6 Change of Variables

Chapter Review Exercises

**Chapter 16: Line and Surface Integrals**

16.1 Vector Fields

16.2 Line Integrals

16.3 Conservative Vector Fields

16.4 Parametrized Surfaces and Surface Integrals

16.5 Surface Integrals of Vector Fields

Chapter Review Exercises

**Chapter 17: Fundamental Theorems of Vector Analysis**

17.1 Green’s Theorem

17.2 Stokes’ Theorem

17.3 Divergence Theorem

Chapter Review Exercises

**Appendices**

A. The Language of Mathematics

B. Properties of Real Numbers

C. Induction and the Binomial Theorem

D. Additional Proofs

Answers to Odd-Numbered Exercises

References

Index

References

Index

## Authors

### Jon Rogawski

**Jon Rogawski** received his undergraduate and master’s degrees in mathematics simultaneously from Yale University, and he earned his PhD in mathematics from Princeton University, where he studied under Robert Langlands. Before joining the Department of Mathematics at UCLA in 1986, where he was a full professor, he held teaching and visiting positions at the Institute for Advanced Study, the University of Bonn, and the University of Paris at Jussieu and Orsay.
Jon’s areas of interest were number theory, automorphic forms, and harmonic analysis on semisimple groups. He published numerous research articles in leading mathematics journals, including the research monograph Automorphic Representations of Unitary Groups in Three Variables (Princeton University Press). He was the recipient of a Sloan Fellowship and an editor of the Pacific Journal of Mathematics and the Transactions of the AMS.
As a successful teacher for more than 30 years, Jon Rogawski listened and learned much from his own students. These valuable lessons made an impact on his thinking, his writing, and his shaping of a calculus text. Sadly, Jon Rogawski passed away in September 2011. Jon’s commitment to presenting the beauty of calculus and the important role it plays in students’ understanding of the wider world is the legacy that lives on in each new edition of Calculus.

**Calculus Early Transcendentals MultiVariable**

Third Edition| 2015

Jon Rogawski

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