12.1 Functions of Two Variables.

12.2 Graphs of Functions of Two Variables.

12.3 Contour Diagrams.

12.4 Linear Functions.

12.5 Functions Of Three Variables.

12.6 Limits and Continuity.

Projects: A Heater in a Room, Light in a Wave-Guide.

13 A FUNDAMENTAL TOOL: VECTORS.

13.1 Displacement Vectors.

13.2 Vectors in General.

13.3 The Dot Product.

13.4 The Cross Product.

Projects: Cross Product of Vectors in the Plane, The Dot Product in Genetics, a Warren Truss.

14 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES.

14.1 The Partial Derivative.

14.2 Computing Partial Derivatives Algebraically.

14.3 Local Linearity and the Differential.

14.4 Gradients and Directional Derivatives in the Plane.

14.5 Gradients and Directional Derivatives in Space.

14.6 The Chain Rule.

14.7 Second-Order Partial Derivatives.

14.8 Differentiability.

Projects: Heat Equation, Matching Birthdays.

15 OPTIMIZATION: LOCAL AND GLOBAL EXTREMA.

15.1 Local Extrema.

15.2 Optimization.

15.3 Constrained Optimization: Lagrange Multipliers.

Projects: Optimization in Manufacturing, Fitting a Line to Data Using Least Squares, Hockey and Entropy.

16 INTEGRATING FUNCTIONS OF SEVERAL VARIABLES.

16.1 The Definite Integral of a Function of Two Variables.

16.2 Iterated Integrals.

16.3 Triple Integrals.

16.4 Double Integrals in Polar Coordinates.

16.5 Integrals in Cylindrical and Spherical Coordinates.

16.6 Applications of Integration to Probability.

16.7 Change of Variables in a Multiple Integral.

Projects: A Connection Between e and π, Average Distance Walked to an Airport Gate.

17 PARAMETERIZATION AND VECTOR FIELDS.

17.1 Parameterized Curves.

17.2 Motion, Velocity, and Acceleration.

17.3 Vector Fields.

17.4 The Flow of a Vector Field.

17.5 Parameterized Surfaces.

Projects: Shooting a Basketball, Kepler’s Second Law, Flux Diagrams.

18 LINE INTEGRALS.

18.1 The Idea of a Line Integral.

18.2 Computing Line Integrals Over Parameterized Curves.

18.3 Gradient Fields and Path-Independent Fields.

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

Projects: Conservation of Energy, Planimeters, Amp`Ere’s Law.

19 FLUX INTEGRALS.

19.1 The Idea of a Flux Integral.

19.2 Flux Integrals For Graphs, Cylinders, and Spheres.

19.3 Flux Integrals Over Parameterized Surfaces.

Projects: Gauss’ Law Applied to a Charged Wire and a Charged Sheet, Flux Across a Cylinder Due to a Point Charge: Obtaining Gauss’ Law from Coulomb’s Law.

20 CALCULUS OF VECTOR FIELDS.

20.1 The Divergence of a Vector Field.

20.2 The Divergence Theorem.

20.3 The Curl of a Vector Field.

20.4 Stokes’ Theorem.

20.5 The Three Fundamental Theorems.

Projects: Divergence of Spherically Symmetric Vector Fields, Divergence of Cylindrically Symmetric Vector Fields.

APPENDIX.

A Roots, Accuracy, and Bounds.

B Complex Numbers.

C Newton’s Method.

D Vectors in the Plane.

E Determinants.