12 Functions to Several Variables 693

12.1 Functions to Two Variables 694

12.2 Graphs and Surfaces 702

12.3 Contour Diagrams 711

12.4 Linear Functions 725

12.5 Functions to Three Variables 732

12.6 Limits and Continuity 739

13 a Fundamental Tool: Vectors 745

13.1 Displacement Vectors 746

13.2 Vectors In General 755

13.3 The Dot Product 763

13.4 The Cross Product 774

14 Differentiating Functions to Several Variables 785

14.1 The Partial Derivative 786

14.2 Computing Partial Derivatives Algebraically 795

14.3 Local Linearity and The Differential 800

14.4 Gradients and Directional Derivatives In The Plane 809

14.5 Gradients and Directional Derivatives In Space 819

14.6 The Chain Rule 827

14.7 Second-Order Partial Derivatives 838

14.8 Differentiability 847

15 Optimization: Local and Global Extrema 855

15.1 Critical Points: Local Extrema and Saddle Points 856

15.2 Optimization 866

15.3 Constrained Optimization: Lagrange Multipliers 876

16 Integrating Functions to Several Variables 889

16.1 The Definite Integral to a Function to Two Variables 890

16.2 Iterated Integrals 898

16.3 Triple Integrals 908

16.4 Double Integrals In Polar Coordinates 916

16.5 Integrals In Cylindrical and Spherical Coordinates 921

16.6 Applications to Integration to Probability 931

17 Parameterization and Vector Fields 937

17.1 Parameterized Curves 938

17.2 Motion, Velocity, and Acceleration 948

17.3 Vector Fields 958

17.4 The Flow to a Vector Field 966

18 Line Integrals 973

18.1 The Idea to a Line Integral 974

18.2 Computing Line Integrals Over Parameterized Curves 984

18.3 Gradient Fields and Path-Independent Fields 992

18.4 Path-Dependent Vector Fields and Green’s Theorem 1003

19 Flux Integrals and Divergence 1017

19.1 The Idea to a Flux Integral 1018

19.2 Flux Integrals For Graphs, Cylinders, and Spheres 1029

19.3 The Divergence to a Vector Field 1039

19.4 The Divergence Theorem 1048

20 The Curl and Stokes’ Theorem 1055

20.1 The Curl to a Vector Field 1056

20.2 Stokes’ Theorem 1064

20.3 The Three Fundamental Theorems 1071

21 Parameters, Coordinates, and Integrals 1077

21.1 Coordinates and Parameterized Surfaces 1078

21.2 Change to Coordinates In a Multiple Integral 1089

21.3 Flux Integrals Over Parameterized Surfaces 1094

Appendices Online

A Roots, Accuracy, and Bounds Online

B Complex Numbers Online

C Newton’s Method Online

D Vectors In The Plane Online

E Determinants Online

Ready Reference 1099

Answers to Odd Numbered Problems 1107

Index 1129

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