e-book

Higher Mathematics for Physics and Engineering [PART 4-6]

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Owing to the rapid advances in the physical sciences and engineering, the demand
for higher-level mathematics is increasing yearly. This book is designed
for advanced undergraduates and graduate students who are interested in the
mathematical aspects of their own fields of study. The reader is assumed to
have a knowledge of undergraduate-level calculus and linear algebra.

There are any number of books available on mathematics for physics and
engineering but they all fall into one of two categories: the one emphasizes
mathematical rigor and the exposition of definitions or theorems, whereas the
other is concerned primarily with applying mathematics to practical problems.
We believe that neither of these approaches alone is particularly helpful
to physicists and engineers who want to understand the mathematical background
of the subjects with which they are concerned. This book is different
in that it provides a short path to higher mathematics via a combination of
these approaches. A sizable portion of this book is devoted to theorems and
definitions with their proofs, and we are convinced that the study of these
proofs, which range from trivial to difficult, is useful for a grasp of the general
idea of mathematical logic. Moreover, several problems have been included at
the end of each section, and complete solutions for all of them are presented
in the greatest possible detail. We firmly believe that ours is a better pedagogical
approach than that found in typical textbooks, where there are many
well-polished problems but no solutions.

This book is essentially self-contained and assumes only standard undergraduate
preparation such as elementary calculus and linear algebra. The
first half of the book covers the following three topics: real analysis, functional
analysis, and complex analysis, along with the preliminaries and four
appendixes. Part I focuses on sequences and series of real numbers of real
functions, with detailed explanations of their convergence properties. We also
emphasize the concepts of Cauchy sequences and the Cauchy criterion that
determine the convergence of infinite real sequences. Part II deals with the
theory of the Hilbert space, which is the most important class of infinite vector
spaces. The completeness property of Hilbert spaces allows one to develop
various types of complex orthonormal polynomials, as described in the middle
of Part II. An introduction to the Lebesgue integration theory, a subject
of ever-increasing importance in physics, is also presented. Part III describes
the theory of complex-valued functions of one complex variable. All relevant
elements including analytic functions, singularity, residue, continuation, and
conformal mapping are described in a self-contained manner. A thorough understanding
of the fundamentals treated is important in order to proceed to
more advanced branches of mathematical physics.

In the second half of the volume, the following three specific topics are
discussed: Fourier analysis, differential equations, and tensor analysis. These
three are the most important subjects in both engineering and the physical
sciences, but their rigorous mathematical structures have hardly been covered
in ordinary textbooks. We know that mathematical rigor is often unnecessary
for practical use. However, the blind usage of mathematical methods as a tool
may lead to a lack of understanding of the symbiotic relationship between
mathematics and the physical sciences. We believe that readers who study
the mathematical structures underlying these three subjects in detail will acquire
a better understanding of the theoretical backgrounds associated with
their own fields. Part IV describes the theory of Fourier series, the Fourier
transform, and the Laplace transform, with a special emphasis on the proofs
of their convergence properties. A more contemporary subject, the wavelet
transform, is also described toward the end of Part IV. Part V deals with ordinary
and partial differential equations. The existence theorem and stability
theory for solutions, which serve as the underlying basis for differential equations,
are described with rigorous proofs. Part VI is devoted to the calculus of
tensors in terms of both Cartesian and non-Cartesian coordinates, along with
the essentials of differential geometry. An alternative tensor theory expressed
in terms of abstract vector spaces is developed toward the end of Part VI.

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Ketersediaan

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Informasi Detail

Judul Seri

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No. Panggil

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Penerbit

New York : Springer-Verlag Berlin Heidelberg., 2010

Deskripsi Fisik

xxi, 688 hlm.

Bahasa

English

ISBN/ISSN

9783540878643

Klasifikasi

530

Tipe Isi

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Tipe Media

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Tipe Pembawa

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Edisi

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Subjek

MATEMATIKA TEKNIK
MATEMATIKA FISIKA

Info Detail Spesifik

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Pernyataan Tanggungjawab

agus

Versi lain/terkait

Lampiran Berkas

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