Geometric Integration Theory
This textbook introduces geometric measure theory through the notion of currents. Currents—continuous linear functionals on spaces of differential forms—are a natural language in which to formulate various types of extremal problems arising in geometry, a
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Advisory Board Anthony W. Knapp, State University of New York at Stony Brook, Emeritus
Steven G. Krantz Harold R. Parks
Geometric Integration Theory
Birkhäuser Boston • Basel • Berlin
Steven G. Krantz American Institute of Mathematics 360 Portage Avenue Palo Alto, CA 94306-2244 U.S.A. [email protected]
Harold R. Parks Department of Mathematics Oregon State University Corvallis, OR 97331 U.S.A. [email protected]
ISBN: 978-0-8176-4676-9 e-ISBN: 978-0-8176-4679-0 DOI: 10.1007/978-0-8176-4679-0 Library of Congress Control Number: 2008922000 Mathematics Subject Classification (2000): Primary: 49-01; Secondary: 26-01, 26B20, 28-01, 28A05, 28A75, 28C05, 28C10, 49Q05, 49Q20, 58C35, 53C65, 58E12 ©2008 Birkhäuser Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 www.birkhauser.com
Dedicated to the memory of Hassler Whitney (1907–1989).
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
1
Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Product Measures and the Fubini–Tonelli Theorem . . . . . . . . 1.4 The Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Generalized Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Hausdorff Distance and Steiner Symmetrization . . . . . . . . . . . . . 1.7 Borel and Suslin Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Carathéodo
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