Advanced Calculus, 3rd Edition

Outlines theory and techniques of calculus, emphasizing strong understanding of concepts, and the basic principles of analysis. Reviews elementary and intermediate calculus and features discussions of elementary-point set theory, and properties of continuous functions. PREFACE 4 CONTENTS 8 Chapter 7 Genral Theorems Of Partial Differentiation 213 7.1 / SUFFICIENT CONDITIONS FOR DIFFERENTIABILITY 214 EXERCISES 216 7.2 / CHANGING THE ORDER OF DIFFERENTIATION 216 7.3 / DIFFERETIALS OF COMPOSITE FUNCTIONS 219 7.4 / THE LAW OF THE MEAN 221 7.5 / TAYLOR'S FORMULA AND SERIES 224 7.6 / SUFFICIENT CONDITIONS FOR A RELATIVE EXTREME 228 MISCELLANEOUS EXERCISES 238 7 / PRELIMINARY REMARKS 213 Chapter 8 / IMPLICIT-FUNTION THEOREMS 239 13 / DOUBLE AND TRIPLE INTEGRALS 393 PRELIMINARY REMARKS 393 13.1 / MOTIVATIONS 393 15 / LINE AND SURFACE INTEGRALS 462 15.1 / POINT FUNCTIONS ON CURVES AND SURFACES 462 15.12 / LINE INTEGRALS 463 15.13 / VECTOR FUNCTIONS AND LINE INTEGRALS WORK 468 15.2 / PARTIAL DERIVATIVES AT THE BOUNDARY OF A REGION 472 15.3 / GREEN'S THEOREM IN THE PLANE 474 CHAPTER 15 ANSWERS 736 Fundamentals Of Elementary Calculus -- The Real Number System -- Continuous Functions -- Extensions Of The Law Of The Mean -- Functions Of Several Variables -- The Elements Of Partial Differentiation -- General Theorems Of Partial Differentiation -- Implicit-function Theorems -- The Inverse Function Theorem With Applications -- Vectors And Vector Fields -- Linear Transformations -- Differential Calculus Of Functions From Rn To Rm -- Double And Triple Integrals -- Curves And Surfaces -- Line And Surface Integrals -- Point-set Theory -- Fundamental Theorems On Continuous Functions -- The Theory Of Integration -- Infinite Series -- Uniform Convergence -- Power Series -- Improper Integrals. Angus E. Taylor, W. Robert Mann. Includes Index.