Student Solutions Manual to accompany Vector Calculus

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

• Vectors, Matrices, and Applications 1 Vectors 1Applications in Geometry and Physics 10The Dot Product 20Matrices and Determinants 30The Cross Product 39Chapter Review 48Calculus of Functions of Several Variables 52Real-Valued and Vector-Valued Functions of Several Variables 52Graph of a Function of Several Variables 62Limits and Continuity 76Derivatives 93Paths and Curves in R[superscript 2]-and R[superscript 3] 112Properties of Derivatives 123Gradient and Directional Derivative 135Cylindrical and Spherical Coordinate Systems 151Chapter Review 159Vector-Valued Functions of One Variable 164World of Curves 164Tangents, Velocity, and Acceleration 181Length of a Curve 191Acceleration and Curvature 200Introduction to Differential Geometry of Curves 209Chapter Review 215Scalar and Vector Fields 219Higher-Order Partial Derivatives 219Taylor's Formula 230Extreme Values of Real-Valued Functions 242Optimization with Constraints and Lagrange Multipliers 261Flow Lines 272Divergence and Curl of a Vector Field 278Implicit Function Theorem 292Appendix: Some Identities of Vector Calculus 298Chapter Review 302Integration Along Paths 306Paths and Parametrizations 306Path Integrals of Real-Valued Functions 316Path Integrals of Vector Fields 325Path Integrals Independent of Path 341Chapter Review 360Double and Triple Integrals 363Double Integrals: Definition and Properties 363Double Integrals Over General Regions 375Examples and Techniques of Evaluation of Double Integrals 394Change of Variables in a Double Integral 401Triple Integrals 417Chapter Review 427Integration Over Surfaces, Properties, and Applications of Integrals 431Parametrized Surfaces 431World of Surfaces 448Surface Integrals of Real-Valued Functions 462Surface Integrals of Vector Fields 474Integrals: Properties and Applications 484Chapter Review 495Classical Integration Theorems of Vector Calculus 499Green's Theorem 499The Divergence Theorem 511Stokes' Theorem 524Differential Forms and Classical Integration Theorems 536Vector Calculus in Electromagnetism 553Vector Calculus in Fluid How 566Chapter Review 576Various Results Used in This Book and Proofs of Differentiation Theorems 581Answers to Odd-Numbered Exercises 590Index 615