Preface1 Remembrance of Things Past1.1 Calculus1.1.1 Limits, Continuity, and the Derivative1.1.2 Properties and Uses of the Derivative1.1.3 The Definite Integral1.1.4 Fundamental Theorem of Calculus1.1.5 Taylor’s Theorem1.2 Linear Algebra1.2.1 Vectors1.2.2 Linear Independence1.2.3 Matrices1.2.4 Eigenvalues and Eigenvectors1.3 Exercises and Projects2 Vector-Valued Functions of One Variable2.1 Curves in the Plane and Space2.2 Limits and Continuity2.3 Derivatives2.4 Velocity, Speed, and Acceleration2.5 Integrals2.6 Applications2.6.1 Projectile Motion2.6.2 Kepler’s Laws of Planetary Motion2.7 Exercises and Projects3 Real-Valued Functions of a Vector: The Derivative3.1 Some Examples3.2 Graphs and Level Sets3.3 Partial Derivatives3.4 Parametrized Surfaces3.5 Applications3.5.1 Utility Functions3.5.2 An Age-Structured Population Model3.6 Exercises and Projects4 Differentiable Functions4.1 Limits and Continuity4.2 Differentiability4.3 Directional Derivatives4.4 A Mean Value Theorem4.5 The Jacobian Matrix4.6 Applications4.6.1 Economic Growth4.6.2 Newton’s Method4.7 Exercises and Projects5 Vector Differential Calculus5.1 Differentiating Compositions of Functions5.1.1 The Little Chain Rule5.1.2 General Chain Rule5.2 Change of Variables5.3 Gradient Fields5.4 Normal Vectors5.5 Implicit Differentiation5.6 Extreme Values 5.6.1 Critical Values 5.6.2 Method of Lagrange Multipliers5.6.3 Second Derivative Criteria5.7 Alternative Coordinate Systems5.7.1 Polar Coordinates5.7.2 Cylindrical Coordinates5.7.3 Spherical Coordinates5.8 Applications5.8.1 Maximizing Utility With Budget Constraint5.8.2 Laplace Equation5.9 Exercises and Projects6 Multiple Integrals 6.1 The Iterated Integral6.2 The Multiple Integral 6.2.1 Definition6.2.2 Existence6.2.3 Double Integrals6.2.4 Triple Integrals 6.3 Properties of the Integral6.4 Jacobians and the Change of Variable6.5 Improper Integrals6.6 Applications6.6.1 Probability6.6.2 Density and Moments6.7 Exercises and Projects7 Calculus Along Curves7.1 Work 7.2 Vector Fields and Line Integrals 7.2.1 Line Integrals7.3 Arc Length and Weighted Curves7.3.1 Weighted Curves7.4 Surfaces of Revolution7.5 Numerical Integration 7.6 Curvature and Normals 7.6.1 Curvature 7.7 Flow Lines and Differential Equations7.7.1 Alternative Notation for Vector Fields 7.8 Applications 7.9 Exercises and Projects8 Vector Field Theory8.1 Divergence and Curl8.1.1 Divergence of a Vector Field8.1.2 Properties of Divergence8.1.3 The Del Operator8.1.4 Curl 8.1.5 Derivative Identities8.2 Conservative Vector Fields8.2.1 Potential Functions8.2.2 Equivalent Notions of Path Equivalence 8.2.3 Symmetric Jacobians and Path Independence 8.3 Green’s Theorem in the Plane8.3.1 Setting for Green’s Theorem8.3.2 A Verification of Green’s Theorem8.3.3 Proof of Green’s Theorem for a Very Simple Region8.3.4 Green’s Theorem for Simple Regions8.3.5 Green’s Theorem for More Complicated Regions8.3.6 Using Green’s Theorem to Evaluate Line Integrals8.3.7 The Divergence Theorem8.3.8 Jacobian Symmetry and Gradient Fields8.3.9 Finding Potentials Using Partial Integration8.4 Surface Integrals 8.4.1 Mass 8.4.2 Integrating Vector Fields Over Surfaces 8.4.3 Orientation 8.5 Gauss’s Theorem 8.5.1 Gauss’s Theorem for Simple Regions8.5.2 Surface Independence 8.5.3 Meaning of Divergence8.6 Stokes’ Theorem 8.6.1 Proving Stokes’ Theorem 8.6.2 Simply Connected Surfaces and Conservative Fields8.6.3 Interpreting Curl 8.6.4 Independence of Surface 8.6.5 An Historical Note 8.7 Applications 8.7.1 Geometry: Shoelace Theorem 8.7.2 Newton’s Law of Gravitational Attraction8.8 Exercises and Projects9 Differential Forms and Vector Calculus9.1 What Are Differential Forms?9.1.1 0-Forms9.1.2 1-Forms9.1.3 Integrating a 1-Form Over a Curve9.1.4 2-Forms 9.1.5 Integrating a 2-Form Over a Surface 9.1.6 3-forms 9.1.7 Integrating a 3-Form Over a Region9.2 Algebra of Forms 9.3 Differentiating Forms9.4 Generalizing the Fundamental Theorem of Calculus 9.4.1 Gauss’ Theorem9.4.2 Stokes’ Theorem 9.5 Exercises and ProjectsIndex
