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- Calculus: Early Transcendentals, Multivariable

# Calculus: Early Transcendentals, Multivariable

## Chapters 10-17Second Edition| ©2012 Jon Rogawski

What’s the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of what they are learning? With your teaching—supported by Rogawski’s

*Calculus*Second Edition—the mo...What’s the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of what they are learning? With your teaching—supported by Rogawski’s

*Calculus*Second Edition—the most successful new calculus text in 25 years!Widely adopted in its first edition, Rogawski’s

Now Rogawski’s

*Calculus*worked for instructors and students by balancing formal precision with a guiding conceptual focus. Rogawski engages students while reinforcing the relevance of calculus to their lives and future studies. Precise mathematics, vivid examples, colorful graphics, intuitive explanations, and extraordinary problem sets all work together to help students grasp a deeper understanding of calculus.Now Rogawski’s

*Calculus*success continues in a meticulously updated new edition. Revised in response to user feedback and classroom experiences, the new edition provides an even smoother teaching and learning experience.

What’s the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of what they are learning? With your teaching—supported by Rogawski’s

*Calculus*Second Edition—the most successful new calculus text in 25 years!Widely adopted in its first edition, Rogawski’s

Now Rogawski’s

*Calculus*worked for instructors and students by balancing formal precision with a guiding conceptual focus. Rogawski engages students while reinforcing the relevance of calculus to their lives and future studies. Precise mathematics, vivid examples, colorful graphics, intuitive explanations, and extraordinary problem sets all work together to help students grasp a deeper understanding of calculus.Now Rogawski’s

*Calculus*success continues in a meticulously updated new edition. Revised in response to user feedback and classroom experiences, the new edition provides an even smoother teaching and learning experience.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

Jon Rogawski’s new edition of *Calculus* builds on the strengths of the bestselling First Edition by incorporating his own classroom experience, as well as extensive feedback from many in the mathematics community, including adopters, nonusers, reviewers, and students. Every section has been carefully revised in order to further polish a text that has been enthusiastically recognized for its meticulous pedagogy and its careful balance among the fundamental pillars of calculus instruction: conceptual understanding, skill development, problem solving, and innovative real world applications.

**Enhanced Exercise Sets—with Approximately 25% New and Revised Problems**

The Second Edition features thousands of new and updated problems. Exercise sets were meticulously reviewed by users and nonusers to assist the author as he revised this cornerstone feature of the text. Rogawski carefully evaluated and rewrote exercise sets as needed to further refine quality, pacing, coverage and quantity.

**New and Larger Variety of Applications**

To show students how calculus directly relates to the real world, the Second Edition features many fresh and creative examples and exercises centered on innovative, contemporary applications from engineering, the life sciences, physical sciences, business, economics, medicine, and the social sciences.

Updated Art Program— with Approximately 15% New Figures

Throughout the Second Edition, there are numerous new and updated figures with refined labeling to increase student understanding. The author takes special care to position the art with the related exposition and provide multiple figures rather than a single one for increased visual support of the concepts.

**Key Content Changes**

Rogawski’s Second Edition includes several content changes in response to feedback from users and reviewers. The key revisions include:

- The topic “Limits at Infinity” has been moved forward from Chapter 4 to Section 2.7 so all types of limits are introduced together (Chapter 2 Limits).

- Coverage of “Differentials” has been expanded in ET Section 4.1 and ET Section 14.4 for those who wish to emphasize differentials in their approach to Linear Approximation.

- “L'Hôpital’s Rule” has been moved up in the ET version so this topic can support the section on graph sketching (Chapter 4 Applications of the Derivative in ET).

- The section on “Numerical Integration” has been moved to the end of the chapter after the techniques of integration are presented (Techniques of Integration chapter).

- A section on “Probability and Integration” was added to allow students to explore new applications of integration important in the physical sciences, as well as in business and the social sciences (Techniques of Integration chapter).

- A new section “Applications of Multiple Integrals” has been added to ET Chapter 15 (LT Chapter 16) to unify the approach to applications of multiple integration and give students an even broader selection of applied problems from the physical and social sciences.

**
Calculus: Early Transcendentals, Multivariable**

Second Edition| ©2012

Jon Rogawski

# Digital Options

**Calculus: Early Transcendentals, Multivariable**

Second Edition| 2012

Jon Rogawski

## Table of Contents

**Chapter 10: Infinite Series**

10.1 Sequences

10.2 Summing an Infinite Series

10.3 Convergence of Series with Positive Terms

10.4 Absolute and Conditional Convergence

10.5 The Ratio and Root Tests

10.6 Power Series

10.7 Taylor Series

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

11.1 Parametric Equations

11.2 Arc Length and Speed

11.3 Polar Coordinates

11.4 Area and Arc Length in Polar Coordinates

11.5 Conic Sections

**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 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 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 15: Multiple Integration**15.1 Integration in 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 Multiplying Integrals

15.6 Change of Variables

**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 17: Fundamental Theorems of Vector Analysis**

17.1 Green’s Theorem

17.2 Stokes’ Theorem

17.3 Divergence Theorem

**Appendices**A. The Language of Mathematics

B. Properties of Real Numbers

C. Mathematical Induction and the Binomial Theorem

D. Additional Proofs of Theorems

E. Taylor Polynomials

**Answers to Odd-Numbered ExercisesReferencesPhoto CreditsIndex**

## 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**

Second Edition| 2012

Jon Rogawski

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