Vector Calculus

Miroslav Lovric, Ph.D, is Associate Professor in the Department of Mathematics and Statistics at McMaster University in Ontario, Canada.

• CHAPTER 1 Vectors, Matrices, and Applications 11.1 Vectors 11.2 Applications in Geometry and Physics 101.3 The Dot Product 201.4 Matrices and Determinants 301.5 The Cross Product 39Chapter Review 48CHAPTER 2 Calculus of Functions of Several Variables 522.1 Real-Valued and Vector-Valued Functions of Several Variables 522.2 Graph of a Function of Several Variables 622.3 Limits and Continuity 762.4 Derivatives 932.5 Paths and Curves in R2 and R3 1122.6 Properties of Derivatives 1232.7 Gradient and Directional Derivative 1352.8 Cylindrical and Spherical Coordinate Systems 151Chapter Review 159CHAPTER 3 Vector-Valued Functions of One Variable 1643.1 World of Curves 1643.2 Tangents, Velocity, and Acceleration 1813.3 Length of a Curve 1913.4 Acceleration and Curvature 2003.5 Introduction to Differential Geometry of Curves 209Chapter Review 215CHAPTER 4 Scalar and Vector Fields 2194.1 Higher-Order Partial Derivatives 2194.2 Taylor’s Formula 2304.3 Extreme Values of Real-Valued Functions 2424.4 Optimization with Constraints and Lagrange Multipliers 2614.5 Flow Lines 2724.6 Divergence and Curl of a Vector Field 2784.7 Implicit Function Theorem 2924.8 Appendix: Some Identities of Vector Calculus 298Chapter Review 302CHAPTER 5 Integration Along Paths 3065.1 Paths and Parametrizations 3065.2 Path Integrals of Real-Valued Functions 3165.3 Path Integrals of Vector Fields 3255.4 Path Integrals Independent of Path 341Chapter Review 360CHAPTER 6 Double and Triple Integrals 3636.1 Double Integrals: Definition and Properties 3636.2 Double Integrals Over General Regions 3756.3 Examples and Techniques of Evaluation of Double Integrals 3946.4 Change of Variables in a Double Integral 4016.5 Triple Integrals 417Chapter Review 427CHAPTER 7 Integration Over Surfaces, Properties, and Applications of Integrals 4317.1 Parametrized Surfaces 4317.2 World of Surfaces 4487.3 Surface Integrals of Real-Valued Functions 4627.4 Surface Integrals of Vector Fields 4747.5 Integrals: Properties and Applications 484Chapter Review 495CHAPTER 8 Classical Integration Theorems of Vector Calculus 4998.1 Green’s Theorem 4998.2 The Divergence Theorem 5118.3 Stokes’ Theorem 5248.4 Differential Forms and Classical Integration Theorems 5368.5 Vector Calculus in Electromagnetism 5538.6 Vector Calculus in Fluid Flow 566Chapter Review 576APPENDIX A Various Results Used in This Book and Proofs of Differentiation Theorems 581APPENDIX B Answers to Odd-Numbered Exercises 590Index 615