# Derivatives and Integrals of Multivariable Functions

Springer Science & Business Media, Aug 22, 2003 - Mathematics - 319 pages
This text is appropriate for a one-semester course in what is usually called ad vanced calculus of several variables. The approach taken here extends elementary results about derivatives and integrals of single-variable functions to functions in several-variable Euclidean space. The elementary material in the single- and several-variable case leads naturally to significant advanced theorems about func tions of multiple variables. In the first three chapters, differentiability and derivatives are defined; prop erties of derivatives reducible to the scalar, real-valued case are discussed; and two results from the vector case, important to the theoretical development of curves and surfaces, are presented. The next three chapters proceed analogously through the development of integration theory. Integrals and integrability are de fined; properties of integrals of scalar functions are discussed; and results about scalar integrals of vector functions are presented. The development of these lat ter theorems, the vector-field theorems, brings together a number of results from other chapters and emphasizes the physical applications of the theory.

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### Contents

 Differentiability of Multivariable Functions ix 12 Derivatives and Partial Derivatives 6 13 The Chain Rule 11 1 4 Higher Derivatives 19 Derivatives of Scalar Functions 29 22 The Mean Value Theorem 33 23 Extreme Values and the Derivative 38 24 Extreme Values and the Second Derivative 43
 43 Domains of Integrability 112 44 Integrability and Sets of Zero Volume 119 Integrals of Scalar Functions 131 52 Properties of Integrals 140 53 Change of Variable 144 54 Generalized Integrals 157 55 Line Integrals 165 56 Surface Integrals 182

 25 Implicit Scalar Functions 48 26 Curves Surfaces Tangents and Normals 56 Derivatives of Vector Functions 69 32 The Inverse Function Theorem 75 33 The Implicit Function Theorem 81 34 Lagranges Method 92 Integrability of Multivariable Functions 101 42 Integrability in a Box 105
 Vector Integrals and the VectorField Theorems 197 62 PathIndependence 206 The Theorems of Green and Stokes 214 64 Gausss Theorem 231 Solutions to Exercises 243 References 309 Index 311 Copyright