Mathematical Analysis of Problems in the Natural Sciences
Vladimir A. Zorich is a distinguished Professor of Mathematics at the University of Moscow who solved the problem of global homeomorphism for space quasi-conformal mappings and provided its far-reaching generalizations. In Mathematical Analysis of Problem
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Vladimir Zorich
Mathematical Analysis of Problems in the Natural Sciences Translated by Gerald Gould
Vladimir Zorich Department of Mathematics (Mech-Math) Moscow State University Vorobievy Gory 119992 Moscow Russia [email protected] Translator Gerald G. Gould School of Mathematics Cardiff University Senghenydd Road CF24 4AG Cardiff UK
Original Russian edition Matematicheskij analiz zadach estestvoznaniya published by MCCME, Moscow, Russia, 2008
ISBN 978-3-642-14812-5 e-ISBN 978-3-642-14813-2 DOI 10.1007/978-3-642-14813-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010938125 Mathematics Subject Classification (2010): 00A69, 00A73, 51Pxx, 76Fxx, 80Axx, 94-xx © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Contents
Part I Analysis of Dimensions of Physical Quantities A few introductory words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Elements of the theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1 Dimension of a physical quantity (preliminaries) . . . . . . . . . . . . 5 1.1.1 Measurement, unit of measurement, measuring process 5 1.1.2 Basic and derived units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Dependent and independent units . . . . . . . . . . . . . . . . . . . 6 1.2 A formula for the dimension of a physical quantity . . . . . . . . . . 6 1.2.1 Change of the numerical values of a physical quantity under a change of the sizes of the basic units. 6 1.2.2 Postulate of the invariance of the ratio of the values of physical quantities with the same name. . . . . . . . . . . . 7 1.2.3 Function of dimension and a formula for the dimension of a physical quantity in a given basis. . . . . . 7 1.3 Fundamental theorem of dimension theory . . . . . . . . . . . . . . . . . 9 1.3.1 The Π-Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 Principle of similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
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Examples of applications . . . . . . . . . . . . . . . . . . . . . .
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