# Multivariable Calculus

CRC Press, Jan 29, 1982 - Mathematics - 546 pages
Classroom-tested and lucidly written, Multivariable Calculus gives a thorough and rigoroustreatment of differential and integral calculus of functions of several variables. Designed as ajunior-level textbook for an advanced calculus course, this book covers a variety of notions,including continuity , differentiation, multiple integrals, line and surface integrals, differentialforms, and infinite series. Numerous exercises and examples throughout the book facilitatethe student's understanding of important concepts.The level of rigor in this textbook is high; virtually every result is accompanied by a proof. Toaccommodate teachers' individual needs, the material is organized so that proofs can be deemphasizedor even omitted. Linear algebra for n-dimensional Euclidean space is developedwhen required for the calculus; for example, linear transformations are discussed for the treatmentof derivatives.Featuring a detailed discussion of differential forms and Stokes' theorem, Multivariable Calculusis an excellent textbook for junior-level advanced calculus courses and it is also usefulfor sophomores who have a strong background in single-variable calculus. A two-year calculussequence or a one-year honor calculus course is required for the most successful use of thistextbook. Students will benefit enormously from this book's systematic approach to mathematicalanalysis, which will ultimately prepare them for more advanced topics in the field.

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### Contents

 SOME PRELIMINARIES 1 The Rudiments of Set Theory 1 Some Logic 7 Mathematical Induction 11 Inequalities and Absolute Value 12 Equivalence Relations 1 5 9 12 15 EUCLIDEAN SPACES AND LINEAR TRANSFORMATIONS 17 The Cartesian Plane 19 Euclidean Spaces 24
 II 209 The Proof of Proposition 2 3 214 THE INVERSE AND IMPLICIT FUNCTION THEOREMS 218 The Inverse Function Theorem 223 The Proof of Theorem 2 2 228 I 230 II 235 INTEGRATION 240

 The Norm in R 28 Linear Transformations between Euclidean Spaces 31 Matrices and Linear Transformations 35 Some Operations on Linear Transformations and Matrices 42 Composites of Linear Transformations and Matrix Multiplication 47 Matrix Multiplication 51 CONTINUOUS FUNCTIONS 58 THE DERIVATIVE 85 THE GEOMETRY OF EUCLIDEAN SPACES 110 HIGHER ORDER DERIVATIVES AND TAYLORS 138 Taylors Theorem for Functions of One Variable 145 Taylors Theorem for Functions of n Variables 154 COMPACT AND CONNECTED SETS 160 Continuous Functions on Compact Sets 169 A Characterization of Compact Sets 172 Uniform Continuity 175 Connected Sets 177 MAXIMA AND MINIMA 182 Quadratic Forms 188 Criteria for Local Maxima and Minima 194 I 199 The Method of Lagrange Multipliers 204
 Properties of the Integral 249 The Integral of a Function of Two Variables 255 The Integral of a Function of n Variables 263 Properties of the Integral 271 The Proof of Theorem 6 2 280 ITERATED INTEGRALS AND THE FUBINI THEOREM 283 Integrals over Nonrectangular Regions 290 More Examples 296 The Proof of Fubinis Theorem 304 Differentiating under the Integral Sign 306 The Change of Variable Formula 309 LINE INTEGRALS 318 SURFACE INTEGRALS 351 DIFFERENTIAL FORMS 380 INTEGRATION OF DIFFERENTIAL FORMS 403 INFINITE SERIES 421 INFINITE SERIES OF FUNCTIONS 443 APPENDIX A Determinants 477 APPENDIX B The Proof of the General Inverse Function Theorem 486 SOLUTIONS TO SELECTED EXERCISES 494 INDEX 517 Copyright