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Vectors, Matrices, and Applications | |
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Vectors | |
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Applications in Geometry and Physics | |
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The Dot Product | |
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Matrices and Determinants | |
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The Cross Product | |
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Chapter Review | |
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Calculus of Functions of Several Variables | |
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Real-Valued and Vector-Valued Functions of Several Variables | |
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Graph of a Function of Several Variables | |
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Limits and Continuity | |
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Derivatives | |
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Paths and Curves in R[superscript 2]-and R[superscript 3] | |
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Properties of Derivatives | |
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Gradient and Directional Derivative | |
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Cylindrical and Spherical Coordinate Systems | |
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Chapter Review | |
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Vector-Valued Functions of One Variable | |
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World of Curves | |
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Tangents, Velocity, and Acceleration | |
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Length of a Curve | |
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Acceleration and Curvature | |
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Introduction to Differential Geometry of Curves | |
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Chapter Review | |
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Scalar and Vector Fields | |
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Higher-Order Partial Derivatives | |
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Taylor's Formula | |
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Extreme Values of Real-Valued Functions | |
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Optimization with Constraints and Lagrange Multipliers | |
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Flow Lines | |
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Divergence and Curl of a Vector Field | |
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Implicit Function Theorem | |
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Appendix: Some Identities of Vector Calculus | |
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Chapter Review | |
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Integration Along Paths | |
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Paths and Parametrizations | |
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Path Integrals of Real-Valued Functions | |
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Path Integrals of Vector Fields | |
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Path Integrals Independent of Path | |
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Chapter Review | |
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Double and Triple Integrals | |
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Double Integrals: Definition and Properties | |
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Double Integrals Over General Regions | |
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Examples and Techniques of Evaluation of Double Integrals | |
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Change of Variables in a Double Integral | |
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Triple Integrals | |
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Chapter Review | |
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Integration Over Surfaces, Properties, and Applications of Integrals | |
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Parametrized Surfaces | |
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World of Surfaces | |
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Surface Integrals of Real-Valued Functions | |
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Surface Integrals of Vector Fields | |
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Integrals: Properties and Applications | |
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Chapter Review | |
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Classical Integration Theorems of Vector Calculus | |
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Green's Theorem | |
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The Divergence Theorem | |
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Stokes' Theorem | |
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Differential Forms and Classical Integration Theorems | |
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Vector Calculus in Electromagnetism | |
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Vector Calculus in Fluid How | |
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Chapter Review | |
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Various Results Used in This Book and Proofs of Differentiation Theorems | |
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Answers to Odd-Numbered Exercises | |
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Index | |