L Y N N H. L 0 0 MIS and S H L 0 M 0 S T ERN B ERG
Department of Mathematics, Harvard University
ADVANCED CALCULUS
REVISED EDITION
JONES AND BARTLETT PUBLISHERS
Boston London
,~"\' ,
~ ",
:,i.; J)
Editorial, Sales, and Customer Service Offices:
Jones and Bartlett Publishers, Inc,
One Exeter Plaza
Boston, MA 02116
Jones and Bartlett Publishers International
POBox 1498
London W6 7RS
England
Copyright © 1990 by Jones and Bartlett Publishers, Inc.
Copyright © 1968 by Addison-Wesley Publishing Company, Inc.
All rights reserved. No part of the material protected by this copyright notice
may be reproduced or utilized in any form, electronic or mechanical, including
photocopying, recording, or by any information storage and retrieval system,
without written permission from the copyright owner.
Printed in the United States of America.
10 9 8 7 6 5 4 3 2
Library of Congress Cataloging-in-Publication Data
Loomis, Lynn H.
Advanced calculus / Lynn H. Loomis and Shlomo Sternberg. -Rev. ed.
p. cm.
Originally published: Reading, Mass. : Addison-Wesley Pub. Co., 1968.
ISBN 0-86720-122-3
1. Calculus. I. Sternberg, Shlomo. II. Title.
QA303.L87 1990
515--dc20 89-15620
CIP
'I
'. ,
, PREFACE
This book is based on an honors course in advanced calculus that we gave in the
1960's. The foundational material, presented in the unstarred sections of Chap-
ters 1 through 11, was normally covered, but different applications of this basic
material were stressed from year to year, and the book therefore contains more
material than was covered in anyone year. It can accordingly be used (with
omissions) as a text for a year's course in advanced calculus, or as a text for a
three-semester introduction to analysis.
These prerequisites are a good grounding in the calculus of one variable
from a mathematically rigorous point of view, together with some acquaintance
with linear algebra. The reader should be familiar with limit and continuity type
arguments and have a certain amount of mathematical sophistication. AB possi-
ble introductory texts, we mention Differential and Integral Calculus by R. Cou-
rant, Calculus by T. Apostol, Calculus by M. Spivak, and Pure Mathematics by
G. Hardy. The reader should also have some experience with partial derivatives.
In overall plan the book divides roughly into a first half which develops the
calculus (principally the differential calculus) in the setting of normed vector
spaces, and a second half which deals with the calculus of differentiable manifolds.
Vector space calculus is treated in two chapters, the differential calculus in
Chapter 3, and the basic theory of ordinary differential equations in Chapter 6.
The other early chapters are auxiliary. The first two chapters develop the neces-
sary purely algebraic theory of vector spaces, Chapter 4 presents the material
on compactness and completeness needed for the more substantive results of
the calculus, and Chapter 5 contains a brief account of the extra structure en-
countered in scalar product spaces. Chapter 7 is devoted to multilinear (tensor)
algebra and is, in the main, a reference chapter for later use. Chapter 8 deals
with the theory of (Riemann) integration on Euclidean spaces and includes (in
exercise form) the fundamental facts about the Fourier transform. Chapters 9
and 10 develop the differential and integral calculus on manifolds, while Chapter
11 treats the exterior calculus of E. Cartan.
The first eleven chapters form a logical unit, each chapter depending on the
results of the preceding chapters. (Of course, many chapters contain material
that can be omitted on first reading; this is generally found in starred sections.)
, On the other hand, Chapters 12, 13, and the latter parts of Chapters 6 and 11
are independent of each other, and are to be regarded as illustrative applications
of the methods developed in the earlier chapters. Presented here are elementary
Sturm-Liouville theory and Fourier series, elementary differential geometry,
potential theory, and classical mechanics. We usually covered only one or two
of these topics in our one-year course.
We have not hesitated to present the same material more than once from
different points of view. For example, although we have selected the contraction
mapping fixed-point theorem as our basic approach to the in1plicit-function
theorem, we have also outlined a "Newton's method" proof in the text and have
sketched still a third proof in the exercises. Similarly, the calculus of variations
is encountered twice-once in the context of the differential calculus of an
infinite-dimensional vector space and later in the context of classical mechanics.
The notion of a submanifold of a vector space is introduced in the early ohapters,
while the invariant definition of a manifold is given later on.
In the introductory treatment of vector space theory, we are more careful
and precise than is customary. In fact, this level of precision of language is not
maintained in the later chapters. Our feeling is that in linear algebra, where the
concepts are so clear and the axioms so familiar, it is pedagogically sound to
illustrate various subtle points, such as distinguishing between spaces that are
normally identified, discussing the naturality of various maps, and so on. Later
on, when overly precise language would be more cumbersome, the reader should
be able to produce for hin1self a more precise version of any assertions that he
finds to be formulated too loosely. Similarly, the proofs in the first few chapters
are presented in more formal detail. Again, the philosophy is that once the
student has mastered the notion of what constitutes a fonnal mathematical
proof, it is safe and more convenient to present arguments in the usual mathe-
matical colloquialisms.
While the level of formality decreases, the level of mathematical sophisti-
cation does not. Thus increasingly abstract and sophisticated mathematical
objects are introduced. It has been our experience that Chapter 9 contains the
concepts most difficult for students to absorb, especially the notions of the
tangent space to a manifold and the Lie derivative of various objects with
respect to a vector field.
Department of Mathematics, Harvard University
ADVANCED CALCULUS
REVISED EDITION
JONES AND BARTLETT PUBLISHERS
Boston London
,~"\' ,
~ ",
:,i.; J)
Editorial, Sales, and Customer Service Offices:
Jones and Bartlett Publishers, Inc,
One Exeter Plaza
Boston, MA 02116
Jones and Bartlett Publishers International
POBox 1498
London W6 7RS
England
Copyright © 1990 by Jones and Bartlett Publishers, Inc.
Copyright © 1968 by Addison-Wesley Publishing Company, Inc.
All rights reserved. No part of the material protected by this copyright notice
may be reproduced or utilized in any form, electronic or mechanical, including
photocopying, recording, or by any information storage and retrieval system,
without written permission from the copyright owner.
Printed in the United States of America.
10 9 8 7 6 5 4 3 2
Library of Congress Cataloging-in-Publication Data
Loomis, Lynn H.
Advanced calculus / Lynn H. Loomis and Shlomo Sternberg. -Rev. ed.
p. cm.
Originally published: Reading, Mass. : Addison-Wesley Pub. Co., 1968.
ISBN 0-86720-122-3
1. Calculus. I. Sternberg, Shlomo. II. Title.
QA303.L87 1990
515--dc20 89-15620
CIP
'I
'. ,
, PREFACE
This book is based on an honors course in advanced calculus that we gave in the
1960's. The foundational material, presented in the unstarred sections of Chap-
ters 1 through 11, was normally covered, but different applications of this basic
material were stressed from year to year, and the book therefore contains more
material than was covered in anyone year. It can accordingly be used (with
omissions) as a text for a year's course in advanced calculus, or as a text for a
three-semester introduction to analysis.
These prerequisites are a good grounding in the calculus of one variable
from a mathematically rigorous point of view, together with some acquaintance
with linear algebra. The reader should be familiar with limit and continuity type
arguments and have a certain amount of mathematical sophistication. AB possi-
ble introductory texts, we mention Differential and Integral Calculus by R. Cou-
rant, Calculus by T. Apostol, Calculus by M. Spivak, and Pure Mathematics by
G. Hardy. The reader should also have some experience with partial derivatives.
In overall plan the book divides roughly into a first half which develops the
calculus (principally the differential calculus) in the setting of normed vector
spaces, and a second half which deals with the calculus of differentiable manifolds.
Vector space calculus is treated in two chapters, the differential calculus in
Chapter 3, and the basic theory of ordinary differential equations in Chapter 6.
The other early chapters are auxiliary. The first two chapters develop the neces-
sary purely algebraic theory of vector spaces, Chapter 4 presents the material
on compactness and completeness needed for the more substantive results of
the calculus, and Chapter 5 contains a brief account of the extra structure en-
countered in scalar product spaces. Chapter 7 is devoted to multilinear (tensor)
algebra and is, in the main, a reference chapter for later use. Chapter 8 deals
with the theory of (Riemann) integration on Euclidean spaces and includes (in
exercise form) the fundamental facts about the Fourier transform. Chapters 9
and 10 develop the differential and integral calculus on manifolds, while Chapter
11 treats the exterior calculus of E. Cartan.
The first eleven chapters form a logical unit, each chapter depending on the
results of the preceding chapters. (Of course, many chapters contain material
that can be omitted on first reading; this is generally found in starred sections.)
, On the other hand, Chapters 12, 13, and the latter parts of Chapters 6 and 11
are independent of each other, and are to be regarded as illustrative applications
of the methods developed in the earlier chapters. Presented here are elementary
Sturm-Liouville theory and Fourier series, elementary differential geometry,
potential theory, and classical mechanics. We usually covered only one or two
of these topics in our one-year course.
We have not hesitated to present the same material more than once from
different points of view. For example, although we have selected the contraction
mapping fixed-point theorem as our basic approach to the in1plicit-function
theorem, we have also outlined a "Newton's method" proof in the text and have
sketched still a third proof in the exercises. Similarly, the calculus of variations
is encountered twice-once in the context of the differential calculus of an
infinite-dimensional vector space and later in the context of classical mechanics.
The notion of a submanifold of a vector space is introduced in the early ohapters,
while the invariant definition of a manifold is given later on.
In the introductory treatment of vector space theory, we are more careful
and precise than is customary. In fact, this level of precision of language is not
maintained in the later chapters. Our feeling is that in linear algebra, where the
concepts are so clear and the axioms so familiar, it is pedagogically sound to
illustrate various subtle points, such as distinguishing between spaces that are
normally identified, discussing the naturality of various maps, and so on. Later
on, when overly precise language would be more cumbersome, the reader should
be able to produce for hin1self a more precise version of any assertions that he
finds to be formulated too loosely. Similarly, the proofs in the first few chapters
are presented in more formal detail. Again, the philosophy is that once the
student has mastered the notion of what constitutes a fonnal mathematical
proof, it is safe and more convenient to present arguments in the usual mathe-
matical colloquialisms.
While the level of formality decreases, the level of mathematical sophisti-
cation does not. Thus increasingly abstract and sophisticated mathematical
objects are introduced. It has been our experience that Chapter 9 contains the
concepts most difficult for students to absorb, especially the notions of the
tangent space to a manifold and the Lie derivative of various objects with
respect to a vector field.
