Differential Forms. Theory and Practice - S.H.Weintraub.pdf

(5371 KB) Pobierz
Differential Forms
Theory and Practice
Differential Forms
Theory and Practice
Second edition
by
Steven H. Weintraub
Lehigh University
Bethlehem, PA, USA
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK
OXFORD • PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE
SYDNEY • TOKYO
Academic Press is an imprint of Elsevier
Academic Press is an imprint of Elsevier
The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK
Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands
225 Wyman Street, Waltham, MA 02451, USA
525 B Street, Suite 1800, San Diego, CA 92101-4495, USA
Second edition 2014
Copyright © 2014 Elsevier Inc. All rights reserved
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any
form or by any means electronic, mechanical, photocopying, recording or otherwise without the
prior written permission of the publisher. Permissions may be sought directly from Elsevier’s
Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax
(+44) (0) 1865 853333; email:
permissions@elsevier.com.
Alternatively you can submit your
request online by visiting the Elsevier web site at
http://elsevier.com/locate/permissions,
and
selecting Obtaining permission to use Elsevier material.
Notice
No responsibility is assumed by the publisher for any injury and/or damage to persons or
property as a matter of products liability, negligence or otherwise, or from any use or operation
of any methods, products, instructions or ideas contained in the material herein. Because of
rapid advances in the medical sciences, in particular, independent verification of diagnoses and
drug dosages should be made.
Library of Congress Cataloging-in-Publication Data
Weintraub, Steven H., author.
Differential forms: theory and practice / by Steven H. Weintraub. -- [2] edition.
pages cm
Includes index.
ISBN 978-0-12-394403-0 (alk. paper)
1. Differential forms. I. Title.
QA381.W45 2014
515’.37--dc23
2013035820
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
For information on all Academic Press publications
visit our web site at
store.elsevier.com
Printed and bound in USA
14 15 16 17 10 9 8 7 6 5 4 3 2 1
ISBN: 978-0-12-394403-0
To my mother and the memory of my father
Preface
Differential forms are a powerful computational and theoretical tool.
They play a central role in mathematics, in such areas as analysis
on manifolds and differential geometry, and in physics as well, in
such areas as electromagnetism and general relativity. In this book,
we present a concrete and careful introduction to differential forms,
at the upper-undergraduate or beginning graduate level, designed with
the needs of both mathematicians and physicists (and other users of
the theory) in mind.
On the one hand, our treatment is concrete. By that we mean that
we present quite a bit of material on how to do computations with
differential forms, so that the reader may effectively use them.
On the other hand, our treatment is careful. By that we mean that
we present precise definitions and rigorous proofs of (almost) all of
the results in this book.
We begin at the beginning, defining differential forms and show-
ing how to manipulate them. First we show how to do algebra with
them, and then we show how to find the exterior derivative
dϕ
of
a differential form
ϕ.
We explain what differential forms really are:
Roughly speaking, a
k-form
is a particular kind of function on
k-tuples
of tangent vectors. (Of course, in order to make sense of
this we must first make sense of tangent vectors.) We carry on to
our main goal, the Generalized Stokes’s Theorem, one of the central
theorems of mathematics. This theorem states:
Theorem
(Generalized
Stokes’s Theorem (GST)). Let M be an
oriented smooth k-manifold with boundary
∂
M (possibly empty) and
let
∂
M be given the induced orientation. Let
ϕ
be a
(k
−
1)-form
on
M with compact support. Then
dϕ
=
M
∂
M
ϕ.

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin