• 1.1 Review of Functions
• 1.2 Representing Functions
• 1.3 Inverse, Exponential, and Logarithmic Functions
• 1.4 Trigonometric Functions and Their Inverses
• Review Exercises

• 2.1 The Idea of Limits
• 2.2 Definitions of Limits
• 2.3 Techniques for Computing Limits
• 2.4 Infinite Limits
• 2.5 Limits at Infinity
• 2.6 Continuity
• 2.7 Precise Definitions of Limits
• Review Exercises

• 3.1 Introducing the Derivative
• 3.2 The Derivative as a Function
• 3.3 Rules of Differentiation
• 3.4 The Product and Quotient Rules
• 3.5 Derivatives of Trigonometric Functions
• 3.6 Derivatives as Rates of Change
• 3.7 The Chain Rule
• 3.8 Implicit Differentiation
• 3.9 Derivatives of Logarithmic and Exponential Functions
• 3.10 Derivatives of Inverse Trigonometric Functions
• 3.11 Related Rates
• Review Exercises

• 4.1 Maxima and Minima
• 4.2 Mean Value Theorem
• 4.3 What Derivatives Tell Us
• 4.4 Graphing Functions
• 4.5 Optimization Problems
• 4.6 Linear Approximation and Differentials
• 4.7 L'Hôpital's Rule
• 4.8 Newton's Method
• 4.9 Antiderivatives
• Review Exercises

• 5.1 Approximating Areas under Curves
• 5.2 Definite Integrals
• 5.3 Fundamental Theorem of Calculus
• 5.4 Working with Integrals
• 5.5 Substitution Rule
• Review Exercises

• 6.1 Velocity and Net Change
• 6.2 Regions Between Curves
• 6.3 Volume by Slicing
• 6.4 Volume by Shells
• 6.5 Length of Curves
• 6.6 Surface Area
• 6.7 Physical Applications
• Review Exercises

• 7.1 Logarithmic and Exponential Functions Revisited
• 7.2 Exponential Models
• 7.3 Hyperbolic Functions
• Review Exercises

• 8.1 Basic Approaches
• 8.2 Integration by Parts
• 8.3 Trigonometric Integrals
• 8.4 Trigonometric Substitutions
• 8.5 Partial Fractions
• 8.6 Integration Strategies
• 8.7 Other Methods of Integration
• 8.8 Numerical Integration
• 8.9 Improper Integrals
• Review Exercises

• 9.1 Basic Ideas
• 9.2 Direction Fields and Euler's Method
• 9.3 Separable Differential Equations
• 9.4 Special First-Order Linear Differential Equations
• 9.5 Modeling with Differential Equations
• Review Exercises

• 10.1 An Overview
• 10.2 Sequences
• 10.3 Infinite Series
• 10.4 The Divergence and Integral Tests
• 10.5 Comparison Tests
• 10.6 Alternating Series
• 10.7 The Ratio and Root Tests
• 10.8 Choosing a Convergence Test
• Review Exercises

• 11.1 Approximating Functions with Polynomials
• 11.2 Properties of Power Series
• 11.3 Taylor Series
• 11.4 Working with Taylor Series
• Review Exercises

• 12.1 Parametric Equations
• 12.2 Polar Coordinates
• 12.3 Calculus in Polar Coordinates
• 12.4 Conic Sections
• Review Exercises

• 13.1 Vectors in the Plane
• 13.2 Vectors in Three Dimensions
• 13.3 Dot Products
• 13.4 Cross Products
• 13.5 Lines and Planes in Space
• 13.6 Cylinders and Quadric Surfaces
• Review Exercises

• 14.1 Vector-Valued Functions
• 14.2 Calculus of Vector-Valued Functions
• 14.3 Motion in Space
• 14.4 Length of Curves
• 14.5 Curvature and Normal Vectors
• Review Exercises

• 15.1 Graphs and Level Curves
• 15.2 Limits and Continuity
• 15.3 Partial Derivatives
• 15.4 The Chain Rule
• 15.5 Directional Derivatives and the Gradient
• 15.6 Tangent Planes and Linear Approximation
• 15.7 Maximum/Minimum Problems
• 15.8 Lagrange Multipliers
• Review Exercises

• 16.1 Double Integrals over Rectangular Regions
• 16.2 Double Integrals over General Regions
• 16.3 Double Integrals in Polar Coordinates
• 16.4 Triple Integrals
• 16.5 Triple Integrals in Cylindrical and Spherical Coordinates
• 16.6 Integrals for Mass Calculations
• 16.7 Change of Variables in Multiple Integrals
• Review Exercises

• 17.1 Vector Fields
• 17.2 Line Integrals
• 17.3 Conservative Vector Fields
• 17.4 Green's Theorem
• 17.5 Divergence and Curl
• 17.6 Surface Integrals
• 17.7 Stokes' Theorem
• 17.8 Divergence Theorem
• Review Exercises

• D2.1 Basic Ideas
• D2.2 Linear Homogeneous Equations
• D2.3 Linear Nonhomogeneous Equations
• D2.4 Applications
• D2.5 Complex Forcing Functions
• Review Exercises

1. Proofs of Selected Theorems
2. Algebra Review ONLINE
3. Complex Numbers ONLINE

• NEW - Video Assignments with corresponding MyLab Math exercises are available for each section of the text. These editable assignments are perfect for any class format—especially online, hybrid, or flipped classroom.

• REVISED - The exercise sets are a major focus of the revision. In response to reviewer and instructor feedback, the authors implemented some significant changes to the exercise sets by rearranging and relabeling exercises, modifying some exercises, and adding many new ones.

• • Of the approximately 10,400 exercises appearing in this edition, 18% are new and many of the exercises from second edition were revised for this edition.

  • A simplified exercise set structure went from including five parts to three: Getting Started, Practice Exercises, Explorations and Challenges.

  • Chapter Review exercises received a major revamp to provide more exercises, particularly intermediate-level problems, and more opportunities for students to choose a strategy of solution. More than 26% of the Chapter Review exercises are new.

• Figures – Based on their experiences with students who learn better once they are able to visualize concepts, the authors devoted considerable time and deliberation to the figures in this book. Whenever possible, the authors let the figures communicate essential ideas using annotations reminiscent of an instructor's voice at the board. Readers will quickly find that the figures facilitate learning in new ways.
• Annotated Examples – Worked-out examples feature annotations (set in blue type) to guide students through the process of a solution and to emphasize that each step in a mathematical argument is rigorously justified. These annotations are designed to echo how instructors talk through examples in lecture. They also provide help for students who may struggle with the algebra and trigonometry steps within the solution process.
• Quick Checks – The narrative is interspersed with Quick Check questions that encourage students to do the calculus as they are reading about it. These questions resemble the kinds of questions instructors pose in class.

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• • UPDATED! Assignable Exercises — The authors analyzed aggregated student usage and performance data from MyLab™ Math for the previous edition of this text. The results of this analysis helped improve the quality and quantity of text and MyLab exercises and learning aids that matter the most to instructors and students.
    
  • Setup & Solve Exercises — These exercises require students to show how they set up a problem as well as the solution, better mirroring what is required on tests. This new exercise type was widely praised by users of the 2nd edition, so more were added to the 3rd edition. Each Setup & Solve Exercise is also now available as a regular question where just the final answer is scored.
    
  • Additional Conceptual Questions focus on deeper, theoretical understanding of the key concepts in calculus. These questions were written by faculty at Cornell University under an NSF grant and are also assignable through Learning Catalytics™.
    
  • Additional practice problems--beyond those in the textbook sections--have been added to selected sections within MyLab Math. These problems are clearly labeled EXTRA and are perfect for use in chapter reviews or practice tests.
    
  • eBook with Interactive Figures — The eBook includes approximately 700 figures that can be manipulated by students to provide a deeper geometric understanding of key concepts and examples as they read and learn new material. The authors have written Interactive Figure Exercises that can be assigned for homework so that students can engage with the figures outside of the classroom.
    
  • NEW! Enhanced Interactive Figures — By incorporating functionality from several standard Interactive Figures, Enhanced Interactive Figures are mathematically richer and ideal for in-class demonstrations. Using a single figure, instructors can illustrate concepts that are difficult for students to visualize and make important connections to key themes of calculus.
    
  • ALL NEW! Instructional Videos — For each section of the text, there is now a new full-lecture video. Many of these videos make use of Interactive Figures to enhance student understanding of concepts. To make it easier for students to navigate to the content they need, each lecture video is segmented into shorter clips (labeled Introduction, Example, or Summary).

• Empower each learner: Each student learns at a different pace. Personalized learning pinpoints the precise areas where each student needs practice, giving all students the support they need—when and where they need it—to be successful.
  
• • NEW! Enhanced Sample Assignments — These section-level assignments include just-in-time prerequisite review, help keep skills fresh with spaced practice of key concepts, and provide opportunities to work exercises without learning aids so students check their understanding.
    
  • An Integrated Review version of the MyLab Math course contains pre-made, assignable quizzes to assess the prerequisite skills needed for each chapter, plus personalized remediation for any gaps in skills that are identified. Each student, therefore, receives just the help that he or she needs--no more, no less.

• Teach your course your way: Your course is unique. So whether you'd like to build your own assignments, teach multiple sections, or set prerequisites, MyLab gives you the flexibility to easily create your course to fit your needs.
  
• • Guided Projects — MyLab Math contains 78 Guided Projects that allow students to work in a directed, step-by-step fashion, with various objectives: to carry out extended calculations, to derive physical models, to explore related theoretical topics, or to investigate new applications of calculus. The Guided Projects vividly demonstrate the breadth of calculus and provide a wealth of mathematical excursions that go beyond the typical classroom experience.
    
  • Learning Catalytics — Now included in all MyLab Math courses, this student response tool uses students' smartphones, tablets, or laptops to engage them in more interactive tasks and thinking during lecture. Learning Catalytics™ fosters student engagement and peer-to-peer learning with real-time analytics. Access pre-built exercises created specifically for calculus, including new Quick Quiz exercises for each section of the text and Additional Conceptual Questions.  

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New to MyLab Math

• Updated Assignable Exercises – The authors analyzed aggregated student usage and performance data from MyLab™ Math for the previous edition of this text. The results of this analysis helped improve the quality and quantity of text and MyLab exercises and learning aids that matter the most to instructors and students.

• Additional practice problems – beyond those in the textbook sections – have been added to selected sections within MyLab Math. These problems are clearly labeled EXTRA and are perfect for use in chapter reviews or practice tests.

• Enhanced Sample Assignments – These section-level assignments include just-in-time prerequisite review, help keep skills fresh with spaced practice of key concepts, and provide opportunities to work exercises without learning aids so students check their understanding. They are assignable and editable within MyLab Math.

• Additional Setup & Solve Exercises – These exercises require students to show how they set up a problem as well as the solution, better mirroring what is required on tests. This new exercise type was widely praised by users of the 2nd edition, so more were added to the 3rd edition.

• Enhanced Interactive Figures – By incorporating functionality from several standard Interactive Figures, Enhanced Interactive Figures are mathematically richer and ideal for in-class demonstrations. Using a single figure, instructors can illustrate concepts that are difficult for students to visualize and make important connections to key themes of calculus.

• Instructional Videos – For each section of the text, there is now a new full-lecture video. Many of these videos make use of Interactive Figures to enhance student understanding of concepts. To make it easier for students to navigate to the content they need, each lecture video is segmented into shorter clips (labeled Introduction, Example, or Summary).

• Quick Quizzes – Each section of the text has Quick Quiz exercises loaded into Learning Catalytics (a student classroom response tool).

• Technology Manuals – These manuals cover Maple 2017, Mathematica 11, and the TI-84 Plus and TI-89. Each manual provides detailed guidance for integrating the software package or graphing calculator throughout the course, including syntax and commands. The projects include instructions and ready-made application files for Maple and Mathematica.

New to the Book

• Video Assignments with corresponding MyLab Math exercises are available for each section of the text. These editable assignments are perfect for any class format—especially online, hybrid, or flipped classroom. 

• The exercise sets are a major focus of the revision. In response to reviewer and instructor feedback, the authors implemented some significant changes to the exercise sets by rearranging and relabeling exercises, modifying some exercises, and adding many new ones.

• • Of the approximately 10,400 exercises appearing in this edition, 18% are new and many of the exercises from second edition were revised for this edition.

  • A simplified exercise set structure went from five parts to three: Getting Started, Practice Exercises, Explorations and Challenges.

  • Chapter Review exercises received a major revamp to provide more exercises, particularly intermediate-level problems, and more opportunities for students to choose a strategy of solution. More than 26% of the Chapter Review exercises are new.

Content Changes

Below are noteworthy changes from the previous edition of the text. Many other detailed changes, not noted here, were made to improve the quality of the narrative and exercises.

Bullet points starting with ** represent major content changes from the previous edition.

Chapter 1: Functions

• Example 1.1.2 was modified with more emphasis on using algebraic techniques to determine the domain and range of a function. To better illustrate a common feature of limits, part (c) was replaced with a rational function that has a common factor in the numerator and denominator.

• Examples 1.1.7 and 1.1.8 from the 2e were moved forward in the narrative so that students get an intuitive feel for the composition of two functions using graphs and tables; compositions of functions using algebraic techniques follow.

• Example 1.1.10, illustrating the importance of secant lines, was made more relevant to students by using real data from a GPS watch during a hike. Corresponding exercises were also added.

• Exercises were added to 1.3 to give students practice at finding inverses of functions using the properties of exponential and logarithmic functions.

• New application exercises (investment problems and a biology problem) were added to 1.3 to further illustrate the usefulness of logarithmic and exponential functions.

Chapter 2: Limits

• Example 2.2.4 was revised, emphasizing an algebraic approach to a function with a jump discontinuity rather than a graphical approach.

• Theorems 2.3 and 2.13 were modified, simplifying the notation to better connect with upcoming material.

• Example 2.3.7 was added to solidify the notions of left, right, and two-sided limits.

• The material explaining the end behavior of exponential and logarithmic functions was reworked, and Example 2.5.6 was added that to show how substitution is used in evaluating limits.

• Exercises were added to 2.5 to illustrate the similarities and differences between limits at infinity and infinite limits. Some easier exercises were included in 2.5 involving limits at infinity of functions containing square roots.

• Example 2.7.5 was added to demonstrate an epsilon-delta proof of a limit of a quadratic function.

• 17 epsilon-delta exercises were added to 2.7 to provide a greater variety of problems involving limits of quadratic, cubic, trigonometric, and absolute value functions.

Chapter 3: Derivatives

• Chapter 3 now begins with a look back at average and instantaneous velocity, first encountered in 2.1, with a corresponding revised example (3.1.1).

• ** The derivative at a point and the derivative as a function are now treated separately in 3.1 and 3.2.

• After defining the derivative at a point in 3.1 with a supporting example, a new subsection was added:  Interpreting the Several exercises were added to 3.3 that require students to use the sum and constant rules, together with geometry, to evaluate derivatives.

• ** The Power Rule for derivatives in 3.4 is stated for all real powers (later proved in 3.9). Example 3.4.4 includes two additional parts to highlight this change, and subsequent examples in upcoming sections rely on the more robust version of the Power Rule. The Power Rule for Rational Exponents in 3.8 was deleted due to this change.

• The intermediate level exercises in 3.4 involving the Product Rule and Quotient Rule are combined together under one unified set of directions.

• ** The derivative of e^x still appears early in the chapter, but the derivative of e^kx is delayed; it appears only after the Chain Rule is introduced in 3.7.

• In 3.7, references to Version 1 and Version 2 of the Chain Rule have been deleted.  Additionally, Chain Rule exercises involving repeated use of the rule were merged with the standard exercises.

• In 3.8, emphasis is added on simplifying derivative formulas for implicitly defined functions; see Examples 3.8.4 and 3.8.5.

• Example 3.11.3 was replaced; the new version shows how similar triangles are used in solving a related-rates problem.

Chapter 4: Applications of the Derivative

• ** The Mean Value Theorem (MVT) was moved from 4.6 to 4.2 so that the proof of Theorem 4.7 is not delayed. Exercises have been added to 4.2 that help students better understand the MVT geometrically, and included where the MVT is used to prove some well-known identities and inequalities.

• Example 4.5.5 was added to give guidance on a certain class of optimization problems.

• Example 4.7.3b was replaced to better drive home the need to simplify after applying l'Hôpital's Rule.

• Most of the intermediate exercises in 4.7 are no longer separated out by the type of indeterminate form, and some problems have been added in which l'Hôpital's Rule does not apply.

• Indefinite integrals of trigonometric functions with argument ax (Table 4.9) were relocated to 5.5 where they are derived with the Substitution Rule. A similar change was made to Table 4.10.

• Example 4.9.7b was added to foreshadow a more complete treatment of the domain of an initial value problem found in Chapter 9.

• A significant number of intermediate antiderivative exercises have been added to 4.9 that require some preliminary work (e.g., factoring, cancellation, expansion) before the antiderivatives can be determined.

Chapter 5: Integration

• Examples 5.1.2 and 5.1.3 on approximating areas were replaced with a friendlier function where the grid points are more transparent; the book returns to these approximations in 5.3 where an exact result is given (Example 5.3.3b).

• Three properties of integrals (bounds on definite integrals) were added in 5.2 (Table 5.5), the last of which is used in the proof of the Fundamental Theorem (5.3).

• Exercises were added to 5.1 and 5.2 where students are required to evaluate Riemann sums using graphs or tables instead of formulas.  These exercises will help students better understand the geometric meaning of Riemann sums.

• More exercises have been added to 5.3 in which the integrand must be simplified first before the integrals can be evaluated.

• A proof of Theorem 5.7 is now offered in 5.5.

• Table 5.6 lists the general integration formulas that were relocated from 4.9 to 5.5; Example 5.5.4 derives these formulas.

Chapter 6 Applications of Integration

Chapter 7: Logarithmic, Exponential, and Hyperbolic Functions

• ** Chapter 6 from the 2e was split into two chapters in order to match the number of chapters in Calculus (Late Transcendentals). The result is a compact Chapter 7.

• Exercises requiring students to evaluate net change using graphs were added to 6.1.

• Exercises in 6.2 involving area calculations with respect to x and y are now combined under one unified set of directions (so that students have to first determine the appropriate variable of integration).

• The number of exercises in 6.3 and 6.4 in which curves are revolved about lines other than the x- and y-axes have been increased. Introductory exercises that step students through the processes used to find volumes have been added.

• A more gentle introduction to lifting problems (specifically, lifting a chain) was added in 6.7 and illustrated in Example 6.7.3, accompanied by additional exercises.

• The introduction to exponential growth (7.2) was rewritten to make a clear distinction between the relative growth rate (or percent change) of a quantity and the rate constant k. The narrative has been revised so that the equation y = y₀e^kt applies to both growth and decay models. This revision resulted in a small change to the half-life formula.

• The variety of applied exercises in 7.2 was increased to further illustrate the utility of calculus in the study of exponential growth and decay.

Chapter 8: Integration Techniques

• Table 8.1 now includes four standard trigonometric integrals that previously appeared in Trigonometric Integrals (8.3); these integrals are derived in Examples 8.1.1 and 8.1.2.

• ** A new section (8.6) was added so that students can master integration techniques (i.e., choose a strategy) apart from the context given in the previous five sections.

• The number and variety of exercises in 8.5 where students must set up the appropriate form of the partial fraction decomposition of a rational function have been increased, including more with irreducible quadratic factors.

• A full derivation of Simpson's Rule was added to 8.8, accompanied by Example 8.8.7, additional figures, and an expanded exercise set.

• ** The Comparison Test for improper integrals was added to 8.9, accompanied by a two-part example (8.9.7). New exercises in 8.9 include some covering doubly infinite improper integrals over infinite intervals.

Chapter 9: Differential Equations

• ** The chapter on differential equations that was only available online in the 2e was converted to a chapter of the text, replacing the single-section coverage found in the 2e.

• More attention was given to the domain of an initial value problem, resulting in the addition and revision of several examples and exercises throughout the chapter.

Chapter 10: Sequences and Infinite Series

• ** The second half of Chapter 10 was reordered: Comparison Tests (10.5), Alternating Series (10.6, which includes the topic of absolute convergence), The Ratio and Root Tests (10.7), and Choosing a Convergence Test (10.8; new section). The 2e section that covered the Comparison, Ratio, and Root Tests was split to avoid overwhelming students with too many tests at one time. Section 10.5 focuses entirely on the comparison tests; 39% of the exercises are new. Alternating Series has been moved before the Ratio and Root Tests so that the latter tests may be stated in their more general form (they now apply to any series rather than only series with positive terms). The final section (10.8) gives students an opportunity to master convergence tests after encountering each of them separately.

• The terminology associated with sequences (10.2) now includes bounded above, bounded below, and bounded (rather than only bounded found in earlier editions).

• Theorem 10.3 (Geometric Sequences) is now developed in the narrative rather than within an example, and an additional example (10.2.3) was added to reinforce the theorem and limit laws from Theorem 10.2.

• Example 10.2.5c uses mathematical induction to find the limit of a sequence defined recursively; this technique is reinforced in the exercise set.

• Example 10.3.3 was replaced with telescoping series that are not geometric and that require re-indexing.

• The number and variety of exercises where the student must determine the appropriate series test necessary to determine convergence of a given series has been increased.

• Properties of Convergent Series (Theorem 10.8) was expanded (two more properties) and moved to 10.3 to better balance the material presented in 10.3 and 10.4. Example 10.3.4 now has two parts to give students more exposure to the theorem.

Chapter 11: Power Series

• Chapter 11 was revised to mesh with the changes made in Chapter 10.

• Some easier intermediate-level exercises have been added to 11.1, where series are estimated using nth partial sums for a given value of n.  

• More exercises have been included in 11.2, where the student must find the radius and interval of convergence.

• Example 11.3.2 was added to illustrate how to choose a different center for a series representation of a function when the original series for the function converges to the function on only part of its domain.

• An issue with the exercises in 11.2 of the previous edition has been addressed by adding more exercises where the intervals of convergence are either closed or contain one, but not both, endpoints.

• An issue with exercises in the previous edition has been addressed by adding many exercises that involve power series centered at locations other than 0.

Chapter 12: Parametric and Polar Curves

• ** The arc length of a two-dimensional curve described by parametric equations was added to 12.1, supported by two examples and additional exercises. Area and surfaces of revolution associated with parametric curves were also added to the exercises.

• Example 12.2.3 derives more general polar coordinate equations for circles.

• The arc length of a curve described in polar coordinates is given in 12.3.

Chapter 13: Vectors and the Geometry of Space

• ** The material from the 2e chapter Vectors and Vector-Valued Functions is now covered in this chapter and the following chapter.

• Example 13.1.5c was added to illustrate how to express a vector as a product of its magnitude and direction.

• The number of applied vector exercises in 13.1 has been increased, starting with some easier exercises, resulting in a better gradation exercises.

• ** A more traditional approach has been adopted to lines and planes; these topics are now covered together in 13.5, followed by cylinders and quadric surfaces in 13.6. This arrangement gives students early exposure to all the basic three-dimensional objects that will be encountered throughout the remainder of the text.

• A discussion of the distance from a point to a line was moved from the exercises into the narrative, supported with Example 13.5.3. Example 13.5.4 finds the point of intersection of two lines. Several related exercises were added to this section.

• There is a larger selection of exercises in 13.6 where the student must identify the quadric surface associated with a given equation. Exercises are also included where students design shapes using quadric surfaces.

Chapter 14: Vector-Valued Functions

• More emphasis was placed on the surface(s) on which a space curve lies in 14.1 and 14.3.

• In 14.1, Exercises have been added where students are asked to find the curve of intersection of two surfaces and where students must verify that a curve lies on a given surface.

• Example 14.3.3c was added to illustrate how a space curve can be mapped onto a sphere.

• Because the arc length of plane curves (described parametrically in 12.1 or with polar coordinates in 12.3) was moved to an earlier location in the text, 14.4 is now a shorter section.

Chapter 15: Functions of Several Variables

• ** Equations of planes and quadric surfaces were removed from this chapter and now appear in Chapter 13.

• The notation in Theorem 15.2 was simplified to match changes made to Theorem 2.3.

• Example 15.4.7 was added to illustrate how the Chain Rule is used to compute second partial derivatives.

• More challenging partial derivative exercises have been added to 15.3 and more challenging Chain Rule exercises to 15.4.

• Example 15.5.7 was expanded to give students more practice finding equations of curves that lie on surfaces.

• Theorem 15.13 was added in 15.5; it's a three-dimensional version of Theorem 15.11.

• Example 15.7.7 was replaced with a more interesting example; the accompanying figure helps tell the story of maximum/minimum problems and can be used to preview Lagrange multipliers.

• Some basic exercises have been added in 15.7 that help students better understand the second derivative test for functions of two variables.

• Example 15.8.1 was modified so that Lagrange multipliers is the clear path to a solution, rather than eliminating one of the variables and using standard techniques. The authors also make it clear that care must be taken when using the method of Lagrange multipliers on sets that are not closed and bounded (absolute maximum and minimum values may not exist).

Chapter 16: Multiple Integration

• Example 16.3.2 was modified because it was too similar to Example 16.3.1.

• More care was given to the notation used with polar, cylindrical, and spherical coordinates (see, for example, Theorem 16.3, and the development of integration in different coordinate systems).

• Example 16.4.3 was modified to make the integration a little more transparent and to show that changing variables to polar coordinates is permissible in more than just the xy-plane.

• More multiple integral exercises were added to 16.1, 16.2, and 16.4, where integration by substitution or integration by parts is needed to evaluate the integrals.

• More exercises have been added to 16.4 in which the integrals must first evaluated with respect to x or y instead of z. Also, more exercises have been added that require triple integrals to be expressed in several orderings.

Chapter 17: Vector Calculus

• The approach to scalar line integrals was streamlined; Example 17.2.1 was modified to reflect this fact.

• Basic exercises have been added in 17.2 emphasizing the geometric meaning of line integrals in a vector field.  A subset of exercises was added where line integrals are grouped so that the student must determine the type of line integral before evaluating the integral.

• Theorem 17.5 was added to 17.3; it addresses the converse of Theorem 17.4. We also promoted the area of a plane region by a line integral to theorem status (Theorem 17.8 in 17.4).

• Example 17.7.3 was replaced to give an example of a surface whose bounding curve is not a plane curve and to provide an example that buttresses the claims made at the end of the section (i.e., Two Final Notes on Stokes' Theorem).

• More line integral exercises were added to 17.3 where the student must first find the potential function before evaluating the line integral over a conservative vector field using the Fundamental Theorem of Line Integrals.

• In 17.7, more challenging surface integrals have been added that are evaluated using Stokes Theorem