1. Vectors 1.1 Vectors in Two and Three Dimensions 1.2 More About Vectors 1.3 The Dot Product 1.4 The Cross Product 1.5 Equations for Planes; Distance Problems 1.6 Some n-dimensional Geometry 1.7 New Coordinate Systems True/False Exercises for Chapter 1 Miscellaneous Exercises for Chapter 1   2. Differentiation in Several Variables 2.1 Functions of Several Variables;Graphing Surfaces 2.2 Limits 2.3 The Derivative 2.4 Properties; Higher-order Partial Derivatives 2.5 The Chain Rule 2.6 Directional Derivatives and the Gradient 2.7 Newton's Method (optional) True/False Exercises for Chapter 2 Miscellaneous Exercises for Chapter 2   3. Vector-Valued Functions 3.1 Parametrized Curves and Kepler's Laws 3.2 Arclength and Differential Geometry 3.3 Vector Fields: An Introduction 3.4 Gradient, Divergence, Curl, and the Del Operator True/False Exercises for Chapter 3 Miscellaneous Exercises for Chapter 3   4. Maxima and Minima in Several Variables 4.1 Differentials and Taylor's Theorem 4.2 Extrema of Functions 4.3 Lagrange Multipliers 4.4 Some Applications of Extrema True/False Exercises for Chapter 4 Miscellaneous Exercises for Chapter 4   5. Multiple Integration 5.1 Introduction: Areas and Volumes 5.2 Double Integrals 5.3 Changing the Order of Integration 5.4 Triple Integrals 5.5 Change of Variables 5.6 Applications of Integration 5.7 Numerical Approximations of Multiple Integrals (optional) True/False Exercises for Chapter 5 Miscellaneous Exercises for Chapter 5   6. Line Integrals 6.1 Scalar and Vector Line Integrals 6.2 Green's Theorem 6.3 Conservative Vector Fields True/False Exercises for Chapter 6 Miscellaneous Exercises for Chapter 6   7. Surface Integrals and Vector Analysis 7.1 Parametrized Surfaces 7.2 Surface Integrals 7.3 Stokes's and Gauss's Theorems 7.4 Further Vector Analysis; Maxwell's Equations True/False Exercises for Chapter 7 Miscellaneous Exercises for Chapter 7   8. Vector Analysis in Higher Dimensions 8.1 An Introduction to Differential Forms 8.2 Manifolds and Integrals of k-forms 8.3 The Generalized Stokes's Theorem True/False Exercises for Chapter 8 Miscellaneous Exercises for Chapter 8   Suggestions for Further Reading Answers to Selected Exercises Index

Introduction of basic linear algebra concepts throughout the text enables multivariable topics to be introduced in more general terms and illuminates the connection between concepts in single- and multivariable calculus. More than 600 diagrams and figures connect analytic work to geometry and assist with visualization. An appropriate level of mathematical rigor collects most of the technical derivations at the ends of sections. Many proofs are available for reference, but positioned so as not to disrupt the flow of main ideas. Large numbers of fully worked examples are integrated throughout to motivate students and to explicate the main ideas and techniques. More than 1,400 exercises are carefully crafted to meet student needs: from practice with the basics, to applications, to mid-level exercises, to more challenging conceptual questions. Also included are optional CAS exercises for instructors who incorporate this technology into the course. Chapter-ending exercises have been carefully crafted to help students synthesize material from multiple sections in the chapter. True/false exercises appear at the end of each chapter (approximately 230 total). Well-chosen advanced topics (such as those in Chapter 8 on Vector Analysis in Higher Dimensions), allow instructors to take the discussion to a higher level—beyond the level of other vector calculus texts.  

210 additional exercises at all levels provide practice for students and increased assignment flexibility for instructors. New, optional sections: Section 2.7 now collects the material on Newton's method for approximating solutions to systems of n equations in n unknowns. This new section is optional. Section 5.7 covers numerical methods for approximating multiple integrals Content updates Section 2.2 contains new proofs of limit properties Section 2.5 contains new proofs of the general multivariable chain rule. Section 4.1 contains new proofs of both single-variable and multivariable versions of Taylor’s theorem. Refinements and clarifications throughout the text including many new and revised examples and explanations PowerPoint® slides and Mathematica® files are now available, showing key concepts and illustrations from the text.