Click here for the highlights of new content features

• Classic Thomas text now available in both standard/traditional and early transcendentals versions.

• Full array of function types and categories available in both versions after a Preliminary Chapter reviewing precalculus treatments. In addition to the usual linear, polynomial, power and rational functions, there are exponential functions, logarithmic functions, parametrizations, inverses of familiar functions, trigonometric functions and inverse trigonometric functions. (The Early Transcendentals version then treats the calculus of these functions all together in the chapters on differentiation and integration.)

• Each section begins with a listing of subsection headings so that key section concepts are readily apparent.

• The text continues to be easy to read, conversational, and directed toward the typical science and engineering calculus student. New mathematical ideas are motivated by easily understood examples, and then reinforced by their applications to real-world problems of immediate interest to the student.

• In this edition, there is an increased emphasis on modeling and applications using real data, and an improved balance of graphical and algebraic/analytic methods and techniques.

• Historical biographies are now expanded on the CD-Rom and Web site. An icon keys the biographies appropriately within the text.

• Specific examples and exercises are highlighted and keyed to a new Web site and CD-Rom technology supplement enhancing visualization of key concepts through computer graphics, video, and student application projects.

• Vectors in the plane are now treated separately in a single chapter at the end of the single-variable calculus material. Three-dimensional vectors are then treated along with the multivariable calculus.

• Exercise sets:

Continue to be grouped with appropriate group headings;

Have titles indicating the content/application for word problems;

Are marked with a technology icon if they require the use of a grapher or computer;

Are grouped into a section titled "Computer Explorations" if a computer or CAS is required.

Volume 1: Single-Variable Calculus

Preliminaries

• Complete treatment of all the familiar precalculus functions.

• Earlier treatment of inverses of familiar functions and parametric equations.

• New section on the importance of modeling in calculus.

• New examples and exercises employing real data and regression analysis using a calculator.

1 Lines

Increments · Slope of a Line · Parallel and Perpendicular Lines · Equations of Lines ·

Applications · Regression Analysis with a Calculator

2 Functions and Graphs

Functions · Domains and Ranges · Viewing and Interpreting Graphs · Increasing versus Decreasing Functions · Even Functions and Odd Functions: Symmetry · Functions Defined in Pieces · How to Shift a Graph · The Absolute Value Function · Composite Functions

3 Exponential Functions

Exponential Growth · Population Growth · The Exponential Function e^x · What Happened to a^x ?

4 Inverse Functions and Logarithms

One-to-One Functions · Inverses · Finding Inverses · Logarithmic Functions · Properties of Logarithms · Applications

5 Trigonometric Functions and Their Inverses

Radian Measure · Graphs of Trigonometric Functions · Values of Trigonometric Functions · Periodicity · Even and Odd Trigonometric Functions · Transformations of Trigonometric Graphs · Identities and the Law of Cosines · Inverse Trigonometric Functions · Identities Involving Arc Sine and Arc Cosine

6. Parametric Equations

Parametrizations of Plane Curves · Lines and Other Curves · Parametrizing Inverse Functions · An Application

7 Modeling Change

Mathematical Models · Simplification · Verifying a Model · A Model Construction Process · Empirical Modeling: Capturing the Trend of Collected Data · Using Calculus in Modeling

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 1 Limits and Continuity

• Limits are introduced by way of rates of change, with a concluding section on tangent lines to connect and complete the initial discussion.

• All the fundamental ideas on limits are now together in a single chapter, including Limits are introduced by way finite limits, infinite limits, asymptotes, limit rules, sin q / q , and so forth.

• Both informal and precise definitions of the limit concept are given (but there is less emphasis on using the precise definition to prove theorems).

1.1 Rates of Change and Limits

Average and Instantaneous Speed · Average Rates of Change and Secant Lines · Limits of Functions · Informal Definition of Limit · Precise Definition of Limit

1.2 Finding Limits and One-Sided Limits

Properties of Limits · Eliminating Zero Denominators Algebraically · Sandwich Theorem · One-Sided Limits · Limits Involving (sin q ) / q

1.3 Limits Involving Infinity

Finite Limits as x ® ± ¥ · Limits of Rational Functions as x ® ± ¥ · Horizontal and Vertical Asymptotes: Infinite Limits · Sandwich Theorem Revisited · Precise Definitions of Infinite Limits · End Behavior Models and Oblique Asymptotes

1.4 Continuity

Continuity at a Point · Continuous Functions · Algebraic Combinations · Composites · Intermediate Value Theorem for Continuous Functions

1.5 Tangent Lines

What Is a Tangent to a Curve? · Finding a Tangent to the Graph of a Function · Rates of Change: Derivative at a Point

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 2 Derivatives

• The derivative as a rate of change is presented earlier to stress its importance in studying motion along a line and in modeling real-world phenomena.

• Derivative rules are in two sections to enhance the clarity and flow of the presentation.

• First and second derivatives for parametric equations are included as an application of the chain rule.

2.1 The Derivative as a Function

Definition of Derivative · Notation · Derivatives of Constants, Powers, Multiples, and Sums · Differentiable on an Interval; One-Sided Derivatives · Graphing f ¢ from Estimated Values · Differentiable Functions are Continuous · Intermediate Value Property of Derivatives · Second- and Higher-Order Derivatives

2.2 The Derivative as a Rate of Change

Instantaneous Rates of Change · Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk · Sensitivity to Change · Derivatives in Economics

2.3 Derivatives of Products, Quotients, and Negative Powers

Products · Quotients · Negative Integer Powers of x

2.4 Derivatives of Trigonometric Functions

Derivative of the Sine Function · Derivative of the Cosine Function · Simple Harmonic Motion · Derivatives of the Other Basic Trigonometric Functions · Continuity of Trigonometric Functions

2.5 The Chain Rule

Derivative of a Composite Function · "Outside-Inside" Rule · Repeated Use of the Chain Rule · Slopes of Parametrized Curves · Power Chain Rule · Melting Ice Cubes

2.6 Implicit Differentiation

Implicitly Defined Functions · Derivatives of Higher Order · Rational Powers of Differentiable Functions

2.7 Related Rates

Related Rate Equations · Solution Strategy

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 3 Applications of Derivatives

• Using the first and second derivatives to determine the shape of a graph is more focused and streamlined.

• New section on using the first and second derivative to produce graphical solutions to autonomous first order differential equations---a graphical prelude to Chapters 4 and 6.

• Sections on optimization, linearization, and Newton’s method retain their previous strengths of interesting motivation for the concepts, and quality application examples and exercises.

3.1 Extreme Values of Functions

The Drilling-Rig Problem · Absolute (Global) Extreme Values · Local (Relative) Extreme Values · Finding Extreme Values

3.2 The Mean Value Theorem and Differential Equations

Rolle’s Theorem · Mean Value Theorem · A Physical Interpretation · Other Consequences · Finding Velocity and Position from Acceleration · Differential Equations and the Height of a Projectile

3.3 The Shape of a Graph

First Derivative Test for Increasing and Decreasing Functions · First Derivative Test for Local Extrema · Concavity · Points of Inflection · Second Derivative Test for Local Extrema · Learning about Functions from Derivatives

3.4 Graphical Solutions of Autonomous Differential Equations

Equilibrium Values and Phase Lines · Stable and Unstable Equilibria · Cooling, A Falling Body Encountering Resistance, and Logistic Growth

3.5 Modeling and Optimization

Examples from Business and Industry · Examples from Mathematics and Physics · Fermat’s Principle and Snell’s Law · Examples from Economics · Modeling Discrete Phenomena with Differentiable Functions

3.6 Linearization and Differentials

Linearization · Differentials · Estimating Change with Differentials · Absolute, Relative, and Percentage Change · Sensitivity to Change · Error in Differential Approximation · Converting Mass to Energy

3.7 Newton’s Method

Procedure for Newton’s Method · The Practice · Convergence Is Usually Assured · But Things Can Go Wrong · Fractal Basins and Newton’s Method

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 4 Integration

• As before, indefinite integrals are presented first, stressing their importance for solving elementary differential equations. The rules for antiderivatives and the substitution method follow next.

• As in the previous edition, estimating with finite sums in a variety of application settings motivates the ideas of Riemann sums and definite integrals. Students see the definite integral early as more than just finding an area.

• The section defining the definite integral as a limit of Riemann sums has been streamlined.

• All of the material on single integral areas (including the area between two curves) is now treated in this chapter.

4.1 Indefinite Integrals, Differential Equations, and Modeling

Finding Antiderivatives- Indefinite Integrals · Initial Value Problems · Mathematical Modeling

4.2 Integral Rules; Integration by Substitution

Rules of Algebra for Antiderivatives · The Integrals of sin² x and cos² x · The Power Rule in Integral Form · Substitution- Running the Chain Rule Backwards

4.3 Estimating with Finite Sums

Area and Cardiac Output · Distance Traveled · Displacement vs. Distance Traveled · Volume of a Sphere · Average Value of a Nonnegative Function · Conclusion

4.4 Riemann Sums and Definite Integrals

Riemann Sums · Terminology and Notation of Integration · Area Under the Graph of a Nonnegative Function · Average Value of a Arbitrary Continuous Function · Properties of Definite Integrals

4.5 The Mean Value and Fundamental Theorems

Mean Value Theorem for Definite Integrals · Fundamental Theorem, Part 1 · A Geometric Interpretation · Fundamental Theorem, Part 2 · Area Connection

4.6 Substitution in Definite Integrals

Substitution Formula · Areas Between Curves · Boundaries with Changing Formulas

4.7 Numerical Integration

Trapezoidal Approximations · Error in the Trapezoidal Approximation · Approximation Using Parabolas · Error in Simpson’s Rule · Which Rule Gives Better Results? · Round-off Errors

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 5 Applications of Integrals

• The treatment on volumes has been combined into two sections.

• Arc length formulas are developed for both (explicit) function and parametric curves in the plane.

• Surface area is moved to Chapter 13 where it is needed for surface integrals. It is now treated in a unified fashion (rather than considering the special case of surfaces of revolution).

• The important applications to springs, pumping and lifting, fluid forces, and moments have all been retained from previous editions.

5.1 Volumes by Slicing and Rotation About an Axis

Volumes by Slicing · Solids of Revolution- Circular Cross Sections · Solids of Revolution- Washer Cross Sections

5.2 Modeling Volume Using Cylindrical Shells

Volume by Cylindrical Shells · The Shell Formula

5.3 Lengths of Plane Curves

A Sine Wave · Length of a Smooth Curve · Dealing with Discontinuities in dy/dx · The Short Differential Formula · Parametric Arc Length Formula

5.4 Springs, Pumping and Lifting

Work Done by a Constant Force · Work Done by a Variable Force Along a Line · Hooke’s Law for Springs: F = kx · Pumping Liquids from Containers

5.5 Fluid Forces

The Constant-Depth Formula for Fluid Force · The Variable-Depth Formula

5.6 Moments and Centers of Mass

Masses Along a Line · Wires and Thin Rods · Masses Distributed over a Plane Region · Thin, Flat Plates · Centroids

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 6 Transcendental Functions and Differential Equations

• This chapter has been reorganized to present immediately the calculus of logarithmic, exponential, and inverse trigonometric functions. (These ideas are presented in Chapters 2 and 3 in the early transcendentals version.)

• Separable variable and linear first order differential equations follow, modeling growth and decay, heat transfer, a body falling with resisting forces, and mixture problems.

• Euler’s method is combined with additional material on population models illustrating graphical, numerical, and analytic solution methods.

6.1 Logarithms

The Natural Logarithm Function · Derivative of y = ln x · The Range of ln x · The Integral ò (1/u) du · The Integrals of tan x and cot x · Logarithmic Differentiation · Derivative of log_a u · Integrals Involving log_a x

6.2 Exponential Functions

Derivatives of Inverses of Differentiable Functions · Another Way to Look at Theorem 1 · The Inverse of ln x and the Number e · The Natural Exponential Function y = e^x · The Derivative and Integral of e^x · The Number e Expressed as a Limit · The General Exponential Function a^x · The Power Rule (Final Form) · The Derivative and Integral of a^u

6.3 Derivatives of Inverse Trigonometric Functions; Integrals

Derivative of the Arcsine · Derivative of the Arctangent · Derivative of the Arcsecant · Derivatives of the Other Three · Integration Formulas

6.4 First Order Separable Differential Equations

General First Order Differential Equations and Solutions · Separable Equations · Slope Fields · Law of Exponential Change · Continuously Compounded Interest · Radioactivity · Heat Transfer: Newton’s Law of Cooling Revisited · Resistance Proportional to Velocity · Torricelli’s Law

6.5 Linear First Order Differential Equations

Linear First Order Equations · Solving the Linear Equation · Mixture Problems · RL Circuits

6.6 Euler’s Method; Population Models

Euler’s Method · Graphical Solutions · Improved Euler’s Method · Exponential Population Model · Logistic Population Model

6.7 Hyperbolic Functions

Definitions and Identities · Derivatives and Integrals · Inverse Hyperbolic Functions · Useful Identities · Derivatives and Integrals

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 7 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals

• Monte Carlo integration is now included with the use of integral tables or Computer Algebra Systems (CAS) to find integrals
• L’Hôpital’s rule is covered in this chapter just prior to its use for calculating some improper integrals and limits of sequences (in Chapter 8)

7.1 Basic Integration Formulas

Algebraic Procedures

7.2 Integration by Parts

Product Rule in Integral Form · Repeated Use · Solving for the Unknown Integral · Tabular Integration

7.3 Partial Fractions

Partial Fractions · General Description of the Method · The Heaviside "Cover-up" Method for Linear Factors · Other Ways to Determine the Coefficients

7.4 Trigonometric Substitutions

Three Basic Substitutions

7.5 Integral Tables, Computer Algebra Systems, and Monte Carlo Integration

Integral Tables · Integration with a CAS · Monte Carlo Numerical Integration

7.6 L’Hôpital’s Rule

Indeterminate Form 0/0 · Indeterminate Forms 8, 8 · 0, 8 - 8 · Indeterminate Forms 1⁸, 0⁰, 8⁰

7.7 Improper Integrals

Infinite Limits of Integration · The Integral ?₁⁸ dx/x^p · Integrands Divergence · Computer Algebra Systems

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 8 Infinite Series

• The basic ideas concerning sequences of numbers and their limits are covered in the first section. The next section (which is optional) treats the more theoretical ideas involving bounded monotonic sequences, and subsequences.
• Most important series convergence tests are presented together in a single, streamlined section.
• Two new (optional) sections at the end of the chapter cover the basics of Fourier series. This inclusion allows for an earlier introduction to these important concepts for students requiring their use in applied science and engineering courses (sometimes long before they take a course in partial differential equations). Completing the elementary introduction to series, these sections illustrate important series representations of functions other than just power series.

8.1 Limits of Sequences of Numbers

Definitions and Notation · Convergence and Divergence · Calculating Limits of Sequences · Using L’Hôpital’s Rule · Limits That Arise Frequently

8.2 Subsequences, Bounded Sequences, and Picard’s Method

Subsequences · Monotonic and Bounded Sequences · Recursively Defined Sequences · Picard’s Method for Finding Roots

8.3 Infinite Series

Series and Partial Sums · Geometric Series · Divergent Series · n^th — Term Test for Divergence · Adding or Deleting Terms · Reindexing · Combining Series

8.4 Series of Nonnegative Terms

Integral Test · Harmonic Series and p-series · Comparison Tests · Ratio and Root Tests

8.5 Alternating Series, Absolute and Conditional Convergence

Alternating Series · Absolute Convergence · Rearranging Series · Procedure for Determining Convergence

8.6 Power Series

Power Series and Convergence · The Radius and Interval of Convergence · Term-by-Term Differentiation · Term-by-Term Integration · Multiplication of Power Series

8.7 Taylor and Maclaurin Series

Constructing a Series · Taylor and Maclaurin Series · Taylor Polynomials · Remainder of a Taylor Polynomial · Estimating the Remainder · Truncation Error · Table of Maclaurin Series · Combining Taylor Series

8.8 Applications of Power Series

Binomial Series for Powers and Roots · Series Solutions of Differential Equations · Evaluating Indeterminate Forms · Arctangents

8.9 Fourier Series

Coefficients in the Fourier Series Expansion · Convergence of the Fourier Series · Periodic Extension

8.10 Fourier Cosine and Sine Series

Even and Odd Functions · Even Extension: Fourier Cosine Series · Odd Extension: Fourier Sine Series · Gibbs Phenomenon

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 9 Vectors in the Plane and Polar Functions

• This constitutes a new chapter on vectors and projectile motion only in the plane. It permits for an earlier self-contained treatment of planar vectors, if desired. The chapter can be covered any time after the coverage of the integral and calculus of exponential and logarithmic functions.
• Chapters P-9 now form a complete package treating the ideas of single variable calculus. (The three-dimensional vector ideas are now presented independently along with the multivariable calculus.)
• Vector ideas are motivated by their application to studying the paths, velocities, accelerations, and forces associated with bodies moving along planar paths.
• The detailed analytic geometry pertaining to conic sections and quadratic equations has been eliminated. These ideas are thoroughly covered in high school and precalculus courses, but we nevertheless review many of the basics as they are needed in earlier chapters.
• Parametrizations of plane curves has been moved to earlier chapters.
• As in previous editions, polar coordinates and the calculus of polar curves are still covered to prepare students for their applications to multivariable calculus.

9.1 Vectors in the Plane

Component Form · Zero Vector and Unit Vectors · Vector Algebra Operations · Standard Unit Vectors · Length and Direction · Tangents and Normals

9.2 Dot Products

Angle Between Vectors · Laws of the Dot Product · Perpendicular (Orthogonal) Vectors · Vector Projections · Work · Writing a Vector as a Sum of Orthogonal Vectors

9.3 Vector-Valued Functions

Planar Curves · Limits and Continuity · Derivatives · Motion · Integrals

9.4 Modeling Projectile Motion

Ideal Projectile Motion · Height, Flight Time, and Range · Ideal Trajectories Are Parabolic · Firing from (x₀, y₀) · Projectile Motion with Wind Gusts

9.5 Polar Coordinates and Graphs

Polar Coordinates · Polar Graphing · Symmetry · Relating Polar and Cartesian Coordinates · Finding Points Where Polar Graphs Intersect

9.6 Calculus of Polar Curves

Slope · Area in the Plane · Length of a Curve

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Volume 2: Multivariable Calculus

(Chapters 8 and 9 are repeated in this volume.)

Chapter 10 Vectors and Motion in Space

• Three-dimensional vectors, the geometry of space, and vector-valued functions defining space curves are now organized together in this single chapter. It now constitutes, and clearly delineates, the entry point for the multivariable calculus.
• Vectors in the plane are reviewed along with the development of the algebra and geometry of 3-D vectors to help students bridge any possible gap between Calculus II and III courses.
• The logical treatment and organization of motion along space curves and the TNB-frame has been retained from the 9^th edition.

10.1 Cartesian (Rectangular) Coordinates and Vectors in Space

Cartesian Coordinates · Vectors in Space · Magnitude · The Zero Vector · Unit Vectors · Length and Direction · Distance and Spheres in Space · Midpoints

10.2 Dot and Cross Products

Dot Products · Properties of the Dot Product · Perpendicular (Orthogonal) Vectors and Projections · The Cross Product of Two Vectors in Space · Properties of the Cross Product · | u x v | is the Area of a Parallelogram · Determinant Formula for u x v · Torque · Triple Scalar or Box Product

10.3 Lines and Planes in Space

Lines and Line Segments in Space · Equations for Planes in Space · Lines of Intersection

10.4 Cylinders and Quadric Surfaces

Cylinders · Quadric Surfaces

10.5 Vector-Valued Functions and Space Curves

Space Curves · Limits and Continuity · Derivatives and Motion · Differentiation Rules · Vector Functions of Constant Length · Integrals of Vector Functions

10.6 Arc Length and the Unit Tangent Vector T

Arc Length Along a Curve · Speed on a Smooth Curve · Unit Tangent Vector T · Curvature and the Principal Unit Normal for Plane Curves · Circle of Curvature and Radius of Curvature

10.7 The TNB Frame; Tangential and Normal Components of Acceleration

Curvature and Normal Vectors for Space Curves · Torsion and the Binormal Vector · Tangential and Normal Components of Acceleration · Formulas for Computing Curvature and Torsion

10.8 Planetary Motion and Satellites

Motion in Polar and Cylindrical Coordinates · Planets Move in Planes · Coordinates and Initial Conditions · Kepler’s First Law (The Conic Section Law) · Kepler’s Second Law (The Equal Area Law) · Proof of Kepler’s First Law · Kepler’s Third Law (The Time-Distance Law) · Orbit Data

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 11 Multivariable Functions and Their Derivatives

• The chapter has been reorganized to improved efficiency and flow. The treatment of partial derivatives with constrained variables had been moved toward the end of the chapter to follow the introduction to Lagrange multipliers. The treatment of linearization and differentials now follows the treatment of directional derivatives, gradient vectors, and tangent planes.
• The treatment of gradients and tangent planes is shorter and more direct.
• A new introduction to extreme values and saddle points compares and contrasts the multivariable case with the single-variable case.
  
• The exercise sets have been streamlined and all applications exercises labeled for quick identification.

11.1 Functions of Several Variables

Function of Two Variables · Domains and Ranges · Graphs and Level Curves of Functions of Two Variables · Contour Curves · Computer Graphing · Functions of Three or More Variables · Level Surfaces of Functions of Three Variables

11.2 Limits and Continuity in Higher Dimensions

Limit of a Function of Two Variables · Continuity of a Function of Two Variables · Functions of More than Two Variables · Continuous Functions Defined on Closed, Bounded Regions

11.3 Partial Derivatives

Partial Derivatives of a Function of Two Variables · Calculations · Functions of More than Two Variables · Partial Derivatives and Continuity · Second-Order Partial Derivatives · The Mixed Derivative Theorem · Partial Derivatives of Still Higher Order · Differentiability

11.4 The Chain Rule

Composite Functions in Higher Dimensions · Functions of Two Variables · Functions of Three Variables · Functions Defined on Surfaces · Implicit Differentiation Revisited · Functions of Many Variables

11.5 Directional Derivatives, Gradient Vectors, and Tangent Planes

Directional Derivatives in the Plane · Interpretation of the Directional Derivative · Calculation · Properties of Directional Derivatives · Gradients and Tangents to Level Curves · Algebra Rules for Gradients · Increments and Distance · Functions of Three Variables · Tangent Planes and Normal Lines · Planes Tangent to a Surface z = f (x, y)

11.6 Linearization and Differentials

Linearization of a Function of Two Variables · Accuracy of the Standard Linear Approximation · Predicting Change with Differentials · Absolute, Relative, and Percentage Change · Functions of More Than Two Variables

11.7 Extreme Values and Saddle Points

Behavior on Closed Bounded Regions · Derivative Tests for Local Extreme Values · Absolute Maxima and Minima on Closed Bounded Regions · Limitations of the First Derivative Test, and Summary

11.8 Lagrange Multipliers

Constrained Maxima and Minima · The Method of Lagrange Multipliers · Lagrange Multipliers with Two Constraints

11.9 Partial Derivatives with Constrained Variables

Decide Which Variables are Dependent and Which are Independent · How to find ?w/?x When the Variables in w = f (x, y, z) are Constrained by Another Equation · Notation · Arrow Diagrams

11.10 Taylor’s Formula for Two Variables

Derivation of the Second Derivative Test · Error Formula for Linear Approximations · Taylor’s Formula for Functions of Two Variables

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 12 Multiple Integrals

• The treatment of the calculation of masses, moments, and centers of mass with multiple integrals is now self-contained. It no longer assumes previous exposure to the single-integral calculations in Chapter 5, which may now be bypassed entirely.
• Again, the practice of titling exercises makes them noticeably easier to select than before.

12.1 Double Integrals

Double Integrals over Rectangles · Properties of Double Integrals · Double Integrals as Volumes · Fubini’s Theorem for Calculating Double Integrals · Double Integrals over Bounded Nonrectangular Regions · Finding Limits of Integration

12.2 Areas, Moments, and Centers of Mass

Areas of Bounded Regions in the Plane · Average Value · Moments and Centers of Mass · Masses Distributed over a Plane Region · Thin, Flat Plates with Continuous Mass Distributions · Moments of Inertia · Centroids of Geometric Figures

12.3 Double Integrals in Polar Form

Integrals in Polar Coordinates · Finding Limits of Integration · Changing Cartesian Integrals into Polar Integrals

12.4 Triple Integrals in Rectangular Coordinates

Triple Integrals · Properties of Triple Integrals · Volume of a Region in Space · Finding Limits of Integration · Average Value of a Function in Space

12.5 Masses and Moments in Three Dimensions

Masses and Moments

12.6 Triple Integrals in Cylindrical and Spherical Coordinates

Integration in Cylindrical Coordinates · Spherical Coordinates · Integration in Spherical Coordinates

12.7 Substitutions in Multiple Integrals

Substitutions in Double Integrals · Substitutions in Triple Integrals

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Chapter 13 Integration in Vector Fields

• In the treatment of Green's Theorem in the plane, circulation density at a point is introduced as the k-component of a more general circulation vector called the curl, to be treated in detail in the later section on Stokes' Theorem. This resolves the apparent inconsistency of having circulation in the plane be represented by a scalar while circulation in space is represented by a vector.

13.1 Line Integrals

Definitions and Notation · Evaluation for Smooth Curves · Additivity · Mass and Moment Calculations

13.2 Vector Fields, Work, Circulation, and Flux

Vector Fields · Gradient Fields · Work Done by a Force Over a Curve in Space · Notation and Evaluation · Flow Integrals and Circulation · Flux Across a Plane Curve

13.3 Path Independence, Potential Functions, and Conservative Fields

Path Independence · Assumptions · Line Integrals in Conservative Fields · Finding Potentials for Conservative Fields · Exact Differential Forms

13.4 Green’s Theorem in the Plane

Flux Density at a Point: Divergence · Circulation Density at a Point: The Curl · Two Forms for Green’s Theorem · Assumptions · Using Green’s Theorem to Evaluate Line Integrals · Proof of Green’s Theorem for Special Regions · Extending the Proof to Other Regions

13.5 Surface Area and Surface Integrals

Surface Area · A Practical Formula · Surface Integrals · Algebraic Properties: The Surface Area Differential · Orientation · Surface Integral for Flux · Moments and Masses of Thin Shells

13.6 Parametrized Surfaces

Parametrizations of Surfaces · Surface Area · Surface Integrals

13.7 Stokes’ Theorem

Circulation Density: Curl· Stokes’ Theorem · Paddle Wheel Interpretation of ?x F · Proof of Stokes’

13.8 Divergence Theorem and a Unified Theory

Divergence in Three Dimensions · Divergence Theorem · Proof of the Divergence Theorem for Special Regions · Divergence Theorem for Other Regions · Gauss’s Law- One of the Four Great Laws of Electromagnetic Theory · Continuity Equation of Hydrodynamics · Unifying the Integral Theorems

Questions to Guide Your Review

Practice Exercises

Additional Exercises

Preliminaries
· Complete treatment of all the familiar precalculus functions.
· Introduction to parametric equations.
· Earlier treatment of inverses of familiar functions, including inverse trigonometric functions.
· Introduction to mathematical modeling, with modeling exercises.
· New examples and exercises employing real data and regression analysis using a calculator.

Chapter 1 Limits and Continuity
· Limits are introduced by way of rates of change, with a concluding section on tangent lines to connect and complete the initial discussion.
· All the fundamental ideas on limits are now together in a single chapter, including finite limits, infinite limits, asymptotes, limit rules, (sin q) / q, and so forth.
· Both informal and precise definitions of the limit concept are given, but there is less emphasis on using the precise definition to prove theorems.

Chapter 2 Derivatives
· The derivative as a rate of change is presented earlier to stress its importance in studying motion along a line in modeling real-world phenomena.
· Differentiation rules are presented in two sections to enhance the clarity and flow of the presentation.
· First and second derivatives for parametric equations are included as an application of the Chain Rule.

Chapter 3 Applications of Derivatives
· The treatment of using the first and second derivatives to determine the shape of a graph is more focused and streamlined.
· There is a new section on using the first and second derivative to produce graphical solutions to autonomous first order differential equations - a graphical prelude to Chapters 4 and 6.
· The new section includes an introduction to population modeling.

Chapter 4 Integration
· As before, indefinite integrals are presented first, stressing their importance for solving elementary differential equations. The rules for antiderivatives and the substitution method follow next.
· As in the previous edition, estimating with finite sums in a variety of application settings motivates the ideas of Riemann sums and definite integrals. Students see the definite integral early as more than just a tool for finding area.
· The section defining the definite integral as a limit of Riemann sums has been streamlined and now focuses on continuous functions. Piecewise-continuous functions are treated in the Additional Exercises at the end of the chapter.
· All the material on single integral area calculations (including areas between curves) is now treated in this chapter.

Chapter 5 Applications of Integrals
· The treatment of volumes has been combined into two sections.
· Arc length formulas are developed for both explicit-function and parametric curves in the plane.
· Surface area has been moved to Chapter 13, where it is needed for surface integrals. There, it is treated in a unified fashion rather than being considered as a special case of surfaces of revolution.
· The important applications to springs, pumping and lifting, fluid forces, and moments have all been retained from the previous edition.

Chapter 6 Transcendental Functions and Differential Equations
· This chapter has been reorganized to immediately present the calculus of logarithmic, exponential, and inverse trigonometric functions. (The differential calculus of these functions is presented in Chapters 2 and 3 in the early transcendentals version.) The treatment includes integrals leading to inverse trigonometric functions.
· Separable variable and linear first order differential equations follow, modeling growth and decay, heat transfer, a falling body with resisting forces, and mixture problems.
· Euler's method and the improved Euler's method are combined with additional material on population models, illustrating graphical, numerical, and analytic solution methods.

Chapter 7 Integration Techniques, L'Hôpital's Rule, and Improper Integrals
· Monte Carlo integration is now included with the use of integral tables or Computer Algebra Systems (CAS) to find integrals.
· L'Hôpital's rule is covered in this chapter just prior to its use for calculating some improper integrals and limits of sequences (in Chapter 8).

Chapter 8 Infinite Series
· The basic ideas concerning sequences of numbers and their limits are covered in the first section. The next section, which is optional, treats the more theoretical ideas involving subsequences and bounded monotonic sequences.
· Most of the important series convergence tests are presented together in a single, streamlined section.
· The new optional sections at the end of the chapter cover the basics of Fourier series. This inclusion allows for an earlier introduction to these important concepts for students requiring their use right away in the applied science and engineering courses. Completing the elementary introduction to series, these sections illustrate important representations of functions by series other than power series.

Chapter 9 Vectors in the Plane and Polar Functions
· This is a new chapter on vectors and projectile motion in the plane, with two sections at the end covering polar coordinates and graphs, and the calculus of polar curves, to prepare students for their use in multivariable calculus. It permits an earlier self-contained treatment of planar vectors, if desired. The chapter can be covered any time after the coverage of the integral and the calculus of exponential and logarithmic functions.
· Chapters P-9 now form a complete package treating the ideas of single variable calculus. Three-dimensional vectors are now presented independently along with multivariable calculus, beginning in Chapter 10.
· Vector ideas are motivated by their application to studying paths, velocities, accelerations, and forces associated with bodies moving along planar paths.
· The detailed analytic geometry of conic sections and quadratic equations has been eliminated. These ideas are thoroughly covered in high school and precalculus courses, but we nevertheless review many of the basics as they are need in earlier chapters.
· Parametrizations of plane curves has been moved to earlier chapters.

Chapter 10 Vectors and Motion in Space
· Three-dimensional vectors, the geometry of space, and vector-valued functions defining space curves are now organized together in this single chapter with fresh introductions and examples. It now constitutes, and clearly delineates, the entry point for the multivariable calculus.
· Letters representing vectors have been changed from upper case to the now more standard lower case.
· Vectors in the plane are reviewed along with the development of the algebra and geometry of three-dimensional vectors to help students bridge any possible gap between Calculus II and III courses.
· The logical treatment and organization of motion along space curves and the TNB-frame has been retained from the previous edition.

Chapter 11 Multivariable Functions and Their Derivatives
· The chapter has been reorganized to improved efficiency and flow. The treatment of partial derivatives with constrained variables had been moved toward the end of the chapter to follow the introduction to Lagrange multipliers. The treatment of linearization and differentials now follows the treatment of directional derivatives, gradient vectors, and tangent planes.
· The treatment of gradients and tangent planes is shorter and more direct.
· A new introduction to extreme values and saddle points compares and contrasts the multivariable case with the single-variable case.
· The exercise sets have been streamlined and all applications exercises labeled for quick identification.

Chapter 12 Multiple Integrals
· The treatment of the calculation of masses, moments, and centers of mass with multiple integrals is now self-contained. It no longer assumes previous exposure to the single-integral calculations in Chapter 5, which may now be bypassed entirely.
· Again, the practice of titling exercises makes them noticeably easier to select than before.

Chapter 13 Integration in Vector Fields
· In the treatment of Green's Theorem in the plane, circulation density at a point is introduced as the k-component of a more general circulation vector called the curl, to be treated in detail in the later section on Stokes' Theorem. This resolves the apparent inconsistency of having circulation in the plane be represented by a scalar while circulation in space is represented by a vector.

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