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eface
   Richard Courant's Differential and Integral Calculus, Vols. I and
II, has been tremendously successful in introducing several gener-
ations of mathematicians to higher mathematics. Throughout, those
volumes presented the important lesson that meaningful mathematics
is created from a union of intuitive imagination and deductive reason-
ing. In preparing this revision the authors have endeavored to main-
tain the healthy balance between these two modes of thinking which
characterized the original work. Although Richard Courant did not
live to see the publication of this revision of Volume II, all major
changes had been agreed upon and drafted by the authors before Dr.
Courant's death in January 1972.
   From the outset, the authors realized that Volume II, which deals
with functions of several variables, would have to be revised more
drastically than Volume I. In particular, it seemed desirable to treat
the fundamental theorems on integration in higher dimensions with
the same degree of rigor and generality applied to integration in one
dimension. In addition, there were a number of new concepts and
topics of basic importance, which, in the opinion of the authors, belong
to an introduction to analysis.
   Only minor changes were made in the short chapters (6, 7, and 8)
dealing, respectively, with Differential Equations, Calculus of Vari-
ations, and Functions of a Complex Variable. In the core of the book,
Chapters 1-5, we retained as much as possible the original scheme of
two roughly parallel developments of each subject at different levels:
an informal introduction based on more intuitive arguments together
with a discussion of applications laying the groundwork for the
subsequent rigorous proofs.
    The material from linear algebra contained in the original Chapter
! seemed inadequate as a foundation for the expanded calculus struc-
ture. Thus, this chapter (now Chapter 2) was completely rewritten and
now presents all the required properties of nth order determinants and
matrices, multilinear forms, Gram determinants, and linear manifolds.
                                                                                                       ..
viii Preface
   The new Chapter 1 contains all the fundamental properties of
linear differential forms and their integrals. These prepare the reader
for the introduction to higher-order exterior differential forms added
to Chapter 3. Also found now in Chapter 3 are a new proof of the
implicit function theorem by successive approximations and a discus-
sion of numbers of critical points and of indices of vector fields in two
dimensions.
   Extensive additions were made to the fundamental properties of
multiple integrals in Chapters 4 and 5. Here one is faced with a familiar
difficulty: integrals over a manifold M, defined easily enough by
subdividing M into convenient pieces, must be shown to be inde-
pendent of the particular subdivision. This is resolved by the sys-
tematic use of the family of Jordan measurable sets with its finite
intersection property and of partitions of unity. In order to minimize
topological complications, only manifolds imbedded smoothly into
Euclidean space are considered. The notion of "orientation" of a
manifold is studied in the detail needed for the discussion of integrals
of exterior differential forms and of their additivity properties. On this
basis, proofs are given for the divergence theorem and for Stokes's
theorem in n dimensions. To the section on Fourier integrals in
Chapter 4 there has been added a discussion of Parseval's identity and
of multiple Fourier integrals.
   Invaluable in the preparation of this book was the continued
generous help extended by two friends of the authors, Professors
Albert A. Blank of Carnegie-Mellon University, and Alan Solomon
of the University of the Negev. Almost every page bears the imprint
of their criticisms, corrections, and suggestions. In addition, they
prepared the problems and exercises for this volume.1
   Thanks are due also to our colleagues, Professors K. O. Friedrichs
and Donald Ludwig for constructive and valuable suggestions, and to
John Wiley and Sons and their editorial staff for their continuing
encourgement  and assistance.
FRITZ JOHN
NewYork
September 1973
 1In contrast to Volume I, these have been incorporated completely into the text;
their solutions can be found at the end of the volume.