Courant & Hilbert - Methods of Mathematical Physics Vol 2
The present volume is concerned with the theory of partial dif-
ferential equations, in particular with parts of this wide field that are
related to concepts of physics and mechanics. Even with this restric-
tion, completeness seems unattainable; to a certain extent the ma-
terial selected corresponds to my personal experience and taste. The
intention is to make an important branch of mathematical analysis
more accessible by emphasizing concepts and methods rather than
presenting a collection of theorems and facts, and by leading from
an elementary level to key points on the frontiers of our knowledge
Almost forty years ago I discussed with David Hilbert the plan of a.
work on mathematical physics. Although Hilbert could not participate
in carrying out the plan, I hope the work, and in particular the present
volume, reflects his scientific ethos, which was always firmly directed
towards the relevant nucleus of a mathematical problem and averse to
merely formal generality. We shall introduce our topics by first con-
centrating on typical specific cases which are suggestive by their
concrete freshness and yet exhibit the core of the underlying abstract
situation. Individual phenomena are not relegated to the role of
special examples; rather, general theories emerge by steps as we reach
higher vantage points from which the details on a "lower level" can
be better viewed, unified, and mastered. Thus, corresponding to the
organic process of learning and teaching, an inductive approach is
favored, sometimes at the expense of the conciseness which can be
gained by a deductive, authoritarian mode of presentation.
This book is essentially self-contained; it corresponds to Volume II
of the German edition of the "Mcthoden der Mathematischen Physik"
which appeared in 1937. The original work was subsequently sup-
pressed by the Ministry of Culture in Nazi Germany; later my loyal
friend Ferdinand Springer was forced out as the head of his famous
publishing house. The reprinting by Interscience Publishers under
license of the United States government (1943) secured the survival of
the book. Ever since, a completely new version in the English language
has been in preparation. During this long period, knowledge in the
field has advanced considerably, and I too have been struggling
towards more comprehensive understanding. Naturally the book
reflects these developments to the extent to which I have shared in
them as an active and as a learning participant.
The table of contents indicates the scope of the present book. It
differs in almost every important detail from the German original.
For example, the theory of characteristics and their role for the theory
of wave propagation is now treated much more adequately than was
possible twenty-five years ago. Also the concept of weak solutions of
differential equations, clarified by Sobolev and Friedrichs and alresly
contained in the German edition, now appears in the context of the
theory of ideal functions which, introduced and called "distributions"
by Laurent Schwartz, have become an indispensable tool of slvanced
calculus. An appendix to Chapter VI contains an elementary pres-
entation of this theory. On the other hand, the material of the last
chapter of the German edition, in particular the discussion of existence
of solutions of elliptic differential equations, did not find room in this
volume. A short third volume on the construction of solutions will
treat these topics, including an account of recent mathematics.
The book as now submitted to the public is certainly uneven in
style, completeness and level of difficulty. Still, I hope that it will be
useful to my fellow students, whether they are beginners, scholars,
mathematicians, other scientists or engineers. Possibly the presence
of various levels in the book might make the terrain all the more
accessible by way of the lower regions.
I am apologetically conscious of the fact that some of the progress
achieved outside of my own sphere may have been inslequately re-
ported or even overlooked in this book. Some of these shortcomings
will be remedied by other publications in the foreseeable future such
as a forthcoming book by GSxding and Leray about their fascinating
work.
The present publication would have been impossible without the
sustained unselfish cooperation given to me by friends. Throughout all
my career I have hsl the rare fortune to work with younger people
who were successively my students, scientific companions and in-
structors. Many of them have long since attained high prominence
and yet have continued their helpful attitude. Kurt O. Friedrichs
and Fritz John, whose scientific association with me began more
than thirty yem's ago, are still actively interested in this work on
mathematical physics.--That this volume is dedicated to I(. 0.
Friedrichs is a natural acknowledgment of a lasting scicntific and
personal friendship.
To the cooperation of Peter D. Lax and Louis Nirenberg I owe
much more than can be expressed by quoting specific details. Peter
Ungar has greatly helped me xvith productive suggestions and criti-
cisms. Also, Lipman Bers has rendered most valuable help and, more-
over, has contributed an important appendix to Chapter IV.
Among younger assistants I must particulm'ly mention Donald
Ludwig whose active and spontaneous pm'ticipation has led to a num-
ber of significant contributions.
Critical revision of parts of the manuscript in different stages was
undertaken by Konrad JSrgens, Herbert Kranzer, Anneli Lax, Hanan
Rubin. Proofs were read by Natascha Brunswick, Susan Hahn,
Reuben Hersh, Alan Jeffrey, Peter Rejto, Brigitte Rellich, Leonard
Sarason, Alan Solomon and others. Jane Richtmyer assisted in pre-
paring the list of ferences and in many other aspects of the produc-
tion. A great deal of the editing was done by Lori Berkowitz.
Most of the technical preparation was in the hands of Ruth MmTay,
who typed and retyped thousands of pages of manuscript, ch'ew the
figre'es and altogether was most instrumental in the exasperating
process of transforming hardly legible ch'afts into the present book.
To all these helpers and to othem, whose names may have been
omitted, ! wish to extend my profound thanks.
Thanks are also due to my patient friend Eric S. Proskauer of Inter-
science.
Finally I wish to thank the Office of Naval Research and the Na-
tional Science Foundation, in particular F. Joachim Weyl and Arthur
Grad, for the effective and understanding support given in the
preparation of this book.
New Rochelle, New York
November 1961
R. COURANT
- Paperback: 560 pages
- Publisher: Wiley-VCH; Volume 1 edition (January 4, 1989)
- Language: English
- ISBN-10: 0471504475
- ISBN-13: 978-0471504474
- Product Dimensions: 6.8 x 1.2 x 9.6 inches
The present volume is concerned with the theory of partial dif-
ferential equations, in particular with parts of this wide field that are
related to concepts of physics and mechanics. Even with this restric-
tion, completeness seems unattainable; to a certain extent the ma-
terial selected corresponds to my personal experience and taste. The
intention is to make an important branch of mathematical analysis
more accessible by emphasizing concepts and methods rather than
presenting a collection of theorems and facts, and by leading from
an elementary level to key points on the frontiers of our knowledge
Almost forty years ago I discussed with David Hilbert the plan of a.
work on mathematical physics. Although Hilbert could not participate
in carrying out the plan, I hope the work, and in particular the present
volume, reflects his scientific ethos, which was always firmly directed
towards the relevant nucleus of a mathematical problem and averse to
merely formal generality. We shall introduce our topics by first con-
centrating on typical specific cases which are suggestive by their
concrete freshness and yet exhibit the core of the underlying abstract
situation. Individual phenomena are not relegated to the role of
special examples; rather, general theories emerge by steps as we reach
higher vantage points from which the details on a "lower level" can
be better viewed, unified, and mastered. Thus, corresponding to the
organic process of learning and teaching, an inductive approach is
favored, sometimes at the expense of the conciseness which can be
gained by a deductive, authoritarian mode of presentation.
This book is essentially self-contained; it corresponds to Volume II
of the German edition of the "Mcthoden der Mathematischen Physik"
which appeared in 1937. The original work was subsequently sup-
pressed by the Ministry of Culture in Nazi Germany; later my loyal
friend Ferdinand Springer was forced out as the head of his famous
publishing house. The reprinting by Interscience Publishers under
license of the United States government (1943) secured the survival of
the book. Ever since, a completely new version in the English language
has been in preparation. During this long period, knowledge in the
field has advanced considerably, and I too have been struggling
towards more comprehensive understanding. Naturally the book
reflects these developments to the extent to which I have shared in
them as an active and as a learning participant.
The table of contents indicates the scope of the present book. It
differs in almost every important detail from the German original.
For example, the theory of characteristics and their role for the theory
of wave propagation is now treated much more adequately than was
possible twenty-five years ago. Also the concept of weak solutions of
differential equations, clarified by Sobolev and Friedrichs and alresly
contained in the German edition, now appears in the context of the
theory of ideal functions which, introduced and called "distributions"
by Laurent Schwartz, have become an indispensable tool of slvanced
calculus. An appendix to Chapter VI contains an elementary pres-
entation of this theory. On the other hand, the material of the last
chapter of the German edition, in particular the discussion of existence
of solutions of elliptic differential equations, did not find room in this
volume. A short third volume on the construction of solutions will
treat these topics, including an account of recent mathematics.
The book as now submitted to the public is certainly uneven in
style, completeness and level of difficulty. Still, I hope that it will be
useful to my fellow students, whether they are beginners, scholars,
mathematicians, other scientists or engineers. Possibly the presence
of various levels in the book might make the terrain all the more
accessible by way of the lower regions.
I am apologetically conscious of the fact that some of the progress
achieved outside of my own sphere may have been inslequately re-
ported or even overlooked in this book. Some of these shortcomings
will be remedied by other publications in the foreseeable future such
as a forthcoming book by GSxding and Leray about their fascinating
work.
The present publication would have been impossible without the
sustained unselfish cooperation given to me by friends. Throughout all
my career I have hsl the rare fortune to work with younger people
who were successively my students, scientific companions and in-
structors. Many of them have long since attained high prominence
and yet have continued their helpful attitude. Kurt O. Friedrichs
and Fritz John, whose scientific association with me began more
than thirty yem's ago, are still actively interested in this work on
mathematical physics.--That this volume is dedicated to I(. 0.
Friedrichs is a natural acknowledgment of a lasting scicntific and
personal friendship.
To the cooperation of Peter D. Lax and Louis Nirenberg I owe
much more than can be expressed by quoting specific details. Peter
Ungar has greatly helped me xvith productive suggestions and criti-
cisms. Also, Lipman Bers has rendered most valuable help and, more-
over, has contributed an important appendix to Chapter IV.
Among younger assistants I must particulm'ly mention Donald
Ludwig whose active and spontaneous pm'ticipation has led to a num-
ber of significant contributions.
Critical revision of parts of the manuscript in different stages was
undertaken by Konrad JSrgens, Herbert Kranzer, Anneli Lax, Hanan
Rubin. Proofs were read by Natascha Brunswick, Susan Hahn,
Reuben Hersh, Alan Jeffrey, Peter Rejto, Brigitte Rellich, Leonard
Sarason, Alan Solomon and others. Jane Richtmyer assisted in pre-
paring the list of ferences and in many other aspects of the produc-
tion. A great deal of the editing was done by Lori Berkowitz.
Most of the technical preparation was in the hands of Ruth MmTay,
who typed and retyped thousands of pages of manuscript, ch'ew the
figre'es and altogether was most instrumental in the exasperating
process of transforming hardly legible ch'afts into the present book.
To all these helpers and to othem, whose names may have been
omitted, ! wish to extend my profound thanks.
Thanks are also due to my patient friend Eric S. Proskauer of Inter-
science.
Finally I wish to thank the Office of Naval Research and the Na-
tional Science Foundation, in particular F. Joachim Weyl and Arthur
Grad, for the effective and understanding support given in the
preparation of this book.
New Rochelle, New York
November 1961
R. COURANT
