PURE AND APPLIED MATHEMATIC* A Senes of Monographs and Textbooks CALCULUS IN VECTOR SPACES Second Edition I Lawrence J. Corwin ] Robert H. Szczarba IN.Cfen. 5|5 <J8t»C 2.ed. Autor: Corwin, Lawrence J. Tltulo: Calculus in vector spaces . 6M541 Ac. 19549 BCM CALCULUS IN VECTOR SPACES PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair N ashed Rutgers University University of Delaware New Brunswick, New Jersey Newark, Delaware CHAIRMEN OF THE EDITORIAL BOARD S. Kobayashi Edwin Hewitt University of California, Berkeley University of Washington Berkeley, California Seattle, Washington EDITORIAL BOARD M. S. Baouendi Donald Passman University of California, University of Wisconsin—Madison San Diego Fred S. Roberts Jane Cronin Rutgers University Rutgers University Gian-Carlo Rota JackK. Hale Massachusetts Institute of Georgia Institute of Technology Technology Marvin Marcus David L. Russell University of California, Virginia Polytechnic Institute Santa Barbara and State University W. S. Massey Walter Schempp Yale University Universitdt Siegen Anil Nerode Mark Teply Cornell University University of Wisconsin—Milwaukee M i I_____. MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1- K. Yano, Integral Formulas in Riemannian Geometry 11970) 2. S. Kobayashi, Hyperbolic Manifolds and Hoiomorphic Mappings (1970) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood, trans.) (1970) 4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation ed.; K. Makowski, trans.) (1971) - ■— 5. L. Narici at a/.. Functional Analysis and Valuation Theory (1971) 6. S. S. Passman, Infinite Group Rings (1971) 7. L. Dornhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1971, 1972) 8. W. Boothby and G. L. Weiss, ads.. Symmetric Spaces (1972) 9. Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, trans.) (1972) 10. L. E. Ward, Jr., Topology (1972) 11. A. Babakhanian, Cohomoiogical Methods in Group Theory (1972) 12. R. Gilmer, Multiplicative Ideal Theory (1972) 13. J. Yeh, Stochastic Processes and the Wiener Integral (1973) 14. J. Barros-Neto, Introduction to the Theory of Distributions (1973) 15. R. Larsen, Functional Analysis (1973) 16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles (1973) 17. C. Procesi, Rings with Polynomial Identities (1973) 18. R. Hermann, Geometry. Physics, and Systems (1973) 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) 20. J. Dieudonne, Introduction to the Theory of Formal Groups (1973) 21. /. Vaisman, Cohomology and Differential Forms (1973) 22. B.-Y. Chen, Geometry of Submanifolds (1973) 23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975) 24. R. Larsen, Banach Algebras (1973) 25. R. O. Kujala and A. L. Vitter, eds.. Value Distribution Theory: Part A; Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973) 26. K. B. Sto/arsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) 28. B. R. McDonald, Finite Rings with Identity (1974) 29. J. Satake, Linear Algebra (S. Koh et al.. trans.) (1975) 30. J. S. Golan, Localization of Noncommutative Rings (1975) 31. G. K/ambauer, Mathematical Analysis (1975) 32. M. K. Agoston, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1976) 34. L. £ Mansfield, Linear Algebra with Geometric Applications (1976) 35. N. J. Pullman, Matrix Theory and Its Applications (1976) 36. B. R. McDonald. Geometric Algebra Over Local Rings (1976) 37. C. W. Groetsch, Generalized Inverses of Linear Operators (1977) 38. J. E. Kuczkowski and J. L. Gersting, Abstract Algebra (1977) 39. C. O. Christenson and W. L. Voxman, Aspects of Topology (1977) 40. M. Nagata, Field Theory (1977) 41. R. L. Long, Algebraic Number Theory (1977) 42. W. F. Pfeifer, Integrals and Measures (1977) 43. R. L. Wheeden and A. Zygmund, Measure and Integral (1977) 44. J. H. Curtiss, Introduction to Functions of a Complex Variable (1978) 45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978) 46. W. S. Massey, Homology and Cohomology Theory (1978) 47. M. Marcus, Introduction to Modern Algebra (1978) 48. £ C. Young. Vector and Tensor Analysis (1978) 49. S. B. Nadler, Jr., Hyperspaces of Sets (1978) 50. S. K. Segal, Topics in Group Kings (1978) 51. A. C. M. van Rooij, Non-Archimedean Functional Analysis (1978) 52. L. Corwin and R. Szczarba, Calculus in Vector Spaces (1979) 53. C. Sadosky, Interpolation of Operators and Singular Integrals (1979) 54. J. Cronin, Differential Equations (1980) 55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980) 56. /. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980) 57. H. /. Freedan, Deterministic Mathematical Models in Population Ecology (1980) 58. S. B. Chae, Lebesgue integration (1980) 59. C. S. Rees et a/.. Theory and Applications of Fourier Analysis (1981) 60. L. Nachbin, introduction to Functional Analysis (R. M. Aron, trans.) (1981) 61. G. Orzech and M. Orzech. Plane Algebraic Curves (1981) 62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis (1981) 63. W. L. Voxman and R. H. Goetschel, Advanced Calculus (1981) 64. L. J. Corwin and R. H. Szczarba, Multivariable Calculus (1982) 65. V. /. /strdfescu, Introduction to Linear Operator Theory (1981) 66. /?. D. Jarvinen, Finite end Infinite Dimensional Linear Spaces (1981) 67. J. K. Beam and P. E. Ehrlich, Global Lorentzien Geometry (1981) 68. D. L. Armacost. The Structure of Locally Compact Abelian Groups (1981) 69. J. W. Brewer and M. K. Smith, eds., Emily Noether: A Tribute (1981) 70. K. H. Kim, Boolean Matrix Theory and Applications (1982) 71. T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982) 72. D. B.Gau/d, Differential Topology (1982) 73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) 74. M. Carme/i, Statistical Theory and Random Matrices (1983) 75. J. H. Carruth et a!.. The Theory of Topologicsl Semigroups (1983) 76. R. L. Faber, Differential Geometry and Relativity Theory (1983) 77. S. Barnett, Polynomials and Linear Control Systems (1983) 78. G. Karpilovsky, Commutative Group Algebras (1983) 79. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1983) 80. /. Vaisman, A First Course in Differential Geometry (1984) 81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) 82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) 83. K. Goebel and S. Reich. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) 84. T. Albu and C. NSstdsescu, Relative Finiteness in Module Theory (1984) 85. K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition (1984) 86. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1984) 87. B. R. McDonald, Linear Algebra Over Commutative Rings (1984) 88. M. Namba, Geometry of Projective Algebraic Curves (1984) 89. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) 90. M. R. Bremner et a!.. Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) 91. A. E. Fekete. Real Linear Algebra (1985) 92. S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985) 93. A. J. Jerri, Introduction to Integral Equations with Applications (1985) 94. G. Karpilovsky, Projective Representations of Finite Groups (1985) 95. L. Narici and E. Beckenstein, Topological Vector Spaces (1985) 96. J. Weeks, The Shape of Space (1985) 97. P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research (1985) 98. J.-A. Chao and W. A. Woyczynski, eds.. Probability Theory and Harmonic Analysis (1986) 99. G. D. Crown et a!.. Abstract Algebra (1986) 100. J. H. Carruth et al.. The Theory of Topological Semigroups, Volume 2 (1986) 101. R. S. Doran and V. A. Be/fi, Characterizations of C-Algebras (1986) 102. M. W. Jeter. Mathematical Programming (1986) ^ 103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (1986) 104. A. Verschoren, Relative Invariants of Sheaves (1987) 105. R. A. Usmani. Applied Linear Algebra (1987) 106. P. Blass and J. Lang. Zariski Surfaces and Differential Equations in Characteristic p > 0 (1987) 107. J. A. Reneke et a!.. Structured Hereditary Systems (1987) _r 108. H. Busemonn and B. B. Phadke, Spaces with Distinguished Geodesics (1987) 109. R. Harte, invertibility and Singularity for Bounded Linear Operators (1988) 110. G. S. Ladde et at.. Oscillation Theory of Differential Equations with Deviating Arguments (1987) 111. L. Dudkin et a/.. Iterative Aggregation Theory (1987) 112. 7. Okubo, Differential Geometry (1987) 113. D. L. Stand and M. L. Stand. Real Analysis with Point-Set Topology (1987) 114. T. C. Gard, Introduction to Stochastic Differential Equations (1988) 115. S. 5. Abbyankar, Enumerative Combinatorics of Young Tableaux (1988) 116. H. Strade and R. Famsteiner, Modular Lie Algebras and Their Representations (1988) 117. J. A. Huckaba, Commutative Rings with Zero Divisors (1988) 118. W. D. Wallis, Combinatorial Designs (1988) 119. W. Wigsfaw, Topological Fields (1988) 120. G. Karpilovsky, Field Theory (1988) 121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1989) 122. W. Kozlowski, Modular Function Spaces (1988) 123. E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (1989) 124. M. Pavel. Fundamentals of Pattern Recognition (1989) 125. V. Laksbmikantham et a/.. Stability Analysis of Nonlinear Systems (1989) 126. R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1989) 127. N. A. Watson. Parabolic Equations on an Infinite Strip (1989) 128. K. J. Hastings, Introduction to the Mathematics of Operations Research (1989) 129. B. Fine, Algebraic Theory of the Bianchi Groups (1989) 130. D. N. Dikranjan et a!.. Topological Groups (1989) 131. J. C. Morgan II, Point Set Theory (1990) 132. P. Biler and A. Witkowski, Problems in Mathematical Analysis (1990) 133. H. J. Sussmann, Nonlinear Controllability and Optimal Control (1990) 134. J.-P. Florens et a/.. Elements of Bayesian Statistics (1990) 135. N. Shell, Topological Fields and Near Valuations (1990) 136. B. F. Doolin and C. F. Martin, Introduction to Differential Geometry for Engineers (1990) 137. S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1990) 138. J. Okniriski, Semigroup Algebras (1990) 139. K. Zhu, Operator Theory in Function Spaces (1990) 140. G. B. Price, An Introduction to Multicomplex Spaces and Functions (1991) 141. R. B. Darst, Introduction to Linear Programming (1991) 142. P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991) 143. 7. Husain, Orthogonal Schauder Bases (1991) 144. J. Foran. Fundamentals of Real Analysis (1991) 145. W. C. Brown, Matrices and Vector Spaces (1991) 146. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces (1991) 147. J. S. Golan and T. Head, Modules and the Structures of Rings (1991) 148. C. Small, Arithmetic of Finite Fields (1991) 149. K. Vang, Complex Algebraic Geometry (1991) 150. D. G. Hoffman et a!.. Coding Theory (1991) 151. M. O. Gonzalez, Classical Complex Analysis (1992) 152. M. 0. Gonzdez, Complex Analysis (1992) 153. L. W. Baggett, Functional Analysis (1992) 154. M. Sniedovich, Dynamic Programming (1992) 155. R. P. Agarwal, Difference Equations and Inequalities (1992) 156. C. Brezinski. Biorthogonality and Its Applications to Numerical Analysis (1992) 157. C. Swartz, An Introduction to Functional Analysis (1992) 158. S. B. Nadler, Jr., Continuum Theory (1992) 159. M. A. Al-Gwaiz, Theory of Distributions (1992) 160. B. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) 161. E. Castillo and M. R. Ruiz-Cobo. Functional Equations and Modelling in Science and Engineering (1992) 162. A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis (1992) 163. A. Charlier et a!.. Tensors and the Clifford Algebra (1992) 164. P. Biter and T. Nadzioja, Problems and Examples in Differential Equations {1992) 165. E. Hansen, Global Optimization Using Interval Analysis {1992) 166. 5. Guerre-Delabriere, Classical Sequences in Banach Spaces (1992) 167. V. C. Wong. Introductory Theory of Topological Vector Spaces (1992) 168. 5. H. Kulkarni and B. V. Umayo, Real Function Algebras (1992) 169. W. C. Brown, Matrices Over Commutative Rings (1993) 170. J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) 171. W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1993) 172. E. C. Young, Vector and Tensor Analysis: Second Edition (1993) 173. T. A. &'clr, Elementary Boundary Value Problems (1993) 174. M. Pavel. Fundamentals of Pattern Recognition: Second Edition (1993) 175. S. A. A/beverio et a!., Noncommutative Distributions (1993) 176. W. Fulks, Complex Variables (1993) 177. M. M. Rao, Conditional Measures and Applications (1993) 178. A. Janicki and A. Weron, Simulation and Chaotic Behavior of o-Stable Stochastic Processes (1994) 179. P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) 180. J. Cronin. Differential Equations: Introduction and Qualitative Theory, Second Edition (1994) 181. S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) 182. X. Mao, Exponential Stability of Stochastic Differential Equations (1994) 183. B. S. Thomson, Symmetric Properties of Real Functions (1994) 184. J. E. Rubio, Optimization and Nonstandard Analysis (1994) 185. J. L. Bueso, P. Jara, and A. Verschoren, Compatibility, Stability, and Sheaves (1995) 186. A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (1995) 187. M. R. Darnel, Theory of Lattice-Ordered Groups (1995) 188. Z. Naniewicz and P. D. Pan agio topoulos, Mathematical Theory of Hemi variational Inequalities and Applications (1995) 189. L. J. Corwin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) 190. L. H. Erbe, Q. Kong, B. G. Zhang. Oscillation Theory for Functional Differential Equations (1995) Additional Volumes in Preparation CALCULUS IN VECTOR SPACES Second Edition Lawrence J. Corwin -t Rutgers University New Brunswick. New Jersey Robert H. Szczarba > Yale University New Haven. Connecticut UFC/BU/BCM 22/01/1998 R688541 Calculus in vector spaces / C389374 515 C86c Marcel Dekker, Inc. New York*Basel• Hong Kong 5 I ' C~' >. A FC—00006046—7 Library of Congress Cataloging-in-Publication Data • r* *-1 •. Corwin, Lawrence J. > Calculus in vector spaces / Lawrence J. Corwin, Robert H. Szczarba. — 2nd ed. p. cm. — (Monographs and textbooks in pure and applied mathematics ; 189) Includes index. ISBN 0-8247-9279-3 (acid-free) 1. Calculus. 2. Vector spaces. I. Szczarba, Robert Henry. II. Title. III. Series...' - .......................... QA303.C823 1994 ! I; ‘ -• • Ui -•!. •. 'V- . •.. : t:•*.! j 515M4—dc20 ...•------------------------j 94-24925 s ! cip The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the address below. This book is printed on acid-free paper. Copyright © 1995 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. a y. Marcel Dekker, Inc. r . 270 Madison Avenue, New York, New York 10016 S f Current printing (last digit): 10 9 8 7 6 5 4 3 2 s f PRINTED IN THE UNITED STATES OF AMERICA \ f Preface to the Second Edition Our primary goal in the first edition of this text was to give a unified, integrated presentation of linear algebra and multivariable calculus. This continues to be our goal in the second edition. In preparing this edition, we have systematically reworked the original, clarifying the language when appropriate, and reorganizing the presentation of some of the material to make it more easily understood. The two most significant instances of reorganization occur in Chapter 5 and Chapter 10. The fundamental notion introduced in Chapter 5 is compactness. The treatment in this edition is intended to get to this notion earlier and to clarify the relationship between compactness as defined in terms of sequences (often called sequential compactness) and compactness defined in terms of coverings. In Chapter 10, we introduce the eigenvectors and eigenvalues and prove several versions of the very important spectral theorem. The presentation here distinguishes between proving the existence of eigenvectors (which is done using elementary max-min theory) and the computation of eigenvalue using determi­ nants. This allows us to state and prove the spectral theorem earlier in the chapter. This text has been used very successfully at Yale in a course designed for freshman who have had a solid background in high school calculus and who are interested in continuing to study mathematics. 1 am pleased to acknowledge helpful feedback from both the students taking the course and the faculty teaching it. In particular, 1 would like to thank Richard Beals, Walter Feit, and George Seligman for their many useful suggestions.