Calculus Multivariable

In Calculus: Multivariable, 12th Edition, an expert team of mathematicians delivers a rigorous and intuitive exploration of calculus, introducing concepts like derivatives and integrals of multivariable functions. Using the Rule of Four, the authors present mathematical concepts from verbal, algebraic, visual, and numerical points of view. The book includes numerous exercises, applications, and examples that help readers learn and retain the concepts discussed within.

PREFACE ix

SUPPLEMENTS x

ACKNOWLEDGMENTS xi

THE ROOTS OF CALCULUS xv

11 Three-Dimensional Space; Vector 657

11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 657

11.2 Vectors 663

11.3 Dot Product; Projections 673

11.4 Cross Product 682

11.5 Parametric Equations of Lines 692

11.6 Planes in 3-Space 698

11.7 Quadric Surfaces 705

11.8 Cylindrical and Spherical Coordinates 715

12 Vector-Valued Functions 723

12.1 Introduction to Vector-Valued Functions 723

12.2 Calculus of Vector-Valued Functions 729

12.3 Change of Parameter; Arc Length 738

12.4 Unit Tangent, Normal, and Binormal Vectors 746

12.5 Curvature 751

12.6 Motion Along a Curve 759

12.7 Kepler's Laws of Planetary Motion 771

13 Partial Derivatives 781

13.1 Functions of Two or More Variables 781

13.2 Limits and Continuity 791

13.3 Partial Derivatives 800

13.4 Differentiability, Differentials, and Local Linearity 812

13.5 The Chain Rule 820

13.6 Directional Derivatives and Gradients 830

13.7 Tangent Planes and Normal Vectors 840

13.8 Maxima and Minima of Functions of Two Variables 845

13.9 Lagrange Multipliers 856

14 Multiple Integrals 866

14.1 Double Integrals 866

14.2 Double Integrals Over Nonrectangular Regions 873

14.3 Double Integrals in Polar Coordinates 882

14.4 Surface Area; Parametric Surfaces 889

14.5 Triple Integrals 902

14.6 Triple Integrals in Cylindrical and Spherical Coordinates 909

14.7 Change of Variables in Multiple Integrals; Jacobians 918

14.8 Centers of Gravity Using Multiple Integrals 930

15 Topics in Vector Calculus 942

15.1 Vector Fields 942

15.2 Line Integrals 951

15.3 Independence of Path; Conservative Vector Fields 966

15.4 Green's Theorem 976

15.5 Surface Integrals 983

15.6 Applications of Surface Integrals; Flux 990

15.7 The Divergence Theorem 999

15.8 Stokes' Theorem 1008

A Appendices

A Trigonometry Review (Summary) App-1

B Functions (Summary) App-8

C New Functions From Old (Summary) App-11

D Families of Functions (Summary) App-16

E Inverse Functions (Summary) App-23

READY REFERENCE RR-1

ANSWERS TO ODD-NUMBERED EXERCISES Ans-1

INDEX Ind-1

Web Appendices (online only)

Available in WileyPLUS

A Trigonometry Review

B Functions

C New Functions From Old

D Families of Functions

E Inverse Functions

F Real Numbers, Intervals, and Inequalities

G Absolute Value

H Coordinate Planes, Lines, and Linear Functions

I Distance, Circles, and Quadratic Equations

J Solving Polynomial Equations

K Graphing Functions Using Calculators and Computer Algebra Systems

L Selected Proofs

M Early Parametric Equations Option

N Mathematical Models

O The Discriminant

P Second-Order Linear Homogeneous Differential Equations

Chapter Web Projects: Expanding the Calculus Horizon (online only)

Available in WileyPLUS

Blammo the Human Cannonball -- Chapter 12

Hurricane Modeling -- Chapter 15

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