Michael Spivak

Michael Spivak Brandeis University Calculus on Manifolds A MODERN APPROACH TO CLASSICAL THEOREMS. OF ADVANCED CALCULUS. ADDISON-WESLEY PUBLISHING COMPANY. The Advanced Book Program Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario Wokingham, England Amsterdam Bonn Sydney Singapore Tokyo Madrid San Juan Paris Seoul Milan Mexico City Taipei Calculus on Manifolds A Modem Approach to Classical Theorems of Advanced Calculus Copyright 1965 by Addison-Wesley Publishing Company All rights reserved Library of Congress Card Catalog Number 66-10910. Manufactured in the United States of America The manuscript was put into production on April 21, 1965;. this volume was published on October 26, 1965. ISBN 0-8053-9021-9. 24 25 26 27 28-CRW-9998979695. Twenty-fourth printing, January 1995. Editors' Foreword Mathematics has been expanding i n all directions at a fabulous rate during the past half century. New fields have emerged, the diffusion into other disciplines has proceeded apace, and our knowledge of the classical areas has grown ever more pro.

history and has undergone a striking metamorphosis. The first statement of the Theorem appears as a postscript to a letter, dated July 2, 18.50, from Sir William Thomson (Lord Kelvin) to Stokes. It appeared publicly as question 8 on the Smith's Prize Examination for 1854. This competitive examination, which was taken annually by the best mathe-

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