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The third of three volumes on partial differential equations, this
is devoted to nonlinear PDE. It treats a number of equations of
classical continuum mechanics, including relativistic versions, as
well as various equations arising in differential geometry, such as
in the study of minimal surfaces, isometric imbedding, conformal
deformation, harmonic maps, and prescribed Gauss curvature. In
addition, some nonlinear diffusion problems are studied. It also
introduces such analytical tools as the theory of L Sobolev spaces,
H lder spaces, Hardy spaces, and Morrey spaces, and also a
development of Calderon-Zygmund theory and paradifferential
operator calculus. The book is aimed at graduate students in
mathematics, and at professional mathematicians with an interest in
partial differential equations, mathematical physics, differential
geometry, harmonic analysis and complex analysis
The first of three volumes on partial differential equations, this
one introduces basic examples arising in continuum mechanics,
electromagnetism, complex analysis and other areas, and develops a
number of tools for their solution, in particular Fourier analysis,
distribution theory, and Sobolev spaces. These tools are then
applied to the treatment of basic problems in linear PDE, including
the Laplace equation, heat equation, and wave equation, as well as
more general elliptic, parabolic, and hyperbolic equations.The book
is targeted at graduate students in mathematics and at professional
mathematicians with an interest in partial differential equations,
mathematical physics, differential geometry, harmonic analysis, and
complex analysis.
This second in the series of three volumes builds upon the basic
theory of linear PDE given in volume 1, and pursues more advanced
topics. Analytical tools introduced here include pseudodifferential
operators, the functional analysis of self-adjoint operators, and
Wiener measure. The book also develops basic differential
geometrical concepts, centred about curvature. Topics covered
include spectral theory of elliptic differential operators, the
theory of scattering of waves by obstacles, index theory for Dirac
operators, and Brownian motion and diffusion.
This text provides an introduction to the theory of partial differential equations. It introduces basic examples of partial differential equations, arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, including particularly Fourier analysis, distribution theory, and Sobolev spaces. These tools are applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. Companion texts, which take the theory of partial differential equations further, are AMS volume 116, treating more advanced topics in linear PDE, and AMS volume 117, treating problems in nonlinear PDE. This book is addressed to graduate students in mathematics and to professional mathematicians, with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.
This is a text for students who have had a three-course calculus
sequence and who are ready to explore the logical structure of
analysis as the backbone of calculus. It begins with a development
of the real numbers, building this system from more basic objects
(natural numbers, integers, rational numbers, Cauchy sequences),
and it produces basic algebraic and metric properties of the real
number line as propositions, rather than axioms. The text also
makes use of the complex numbers and incorporates this into the
development of differential and integral calculus. For example, it
develops the theory of the exponential function for both real and
complex arguments, and it makes a geometrical study of the
curve (expit)(expit), for real tt, leading to a
self-contained development of the trigonometric functions and to a
derivation of the Euler identity that is very different from what
one typically sees. Further topics include metric spaces, the
Stone–Weierstrass theorem, and Fourier series.
This text develops linear algebra with the view that it is an
important gateway connecting elementary mathematics to more
advanced subjects, such as advanced calculus, systems of
differential equations, differential geometry, and group
representations. The purpose of this book is to provide a treatment
of this subject in sufficient depth to prepare the reader to tackle
such further material. The text starts with vector spaces, over the
sets of real and complex numbers, and linear transformations
between such vector spaces. Later on, this setting is extended to
general fields. The reader will be in a position to appreciate the
early material on this more general level with minimal effort.
Notable features of the text include a treatment of determinants,
which is cleaner than one often sees, and a high degree of contact
with geometry and analysis, particularly in the chapter on linear
algebra on inner product spaces. In addition to studying linear
algebra over general fields, the text has a chapter on linear
algebra over rings. There is also a chapter on special structures,
such as quaternions, Clifford algebras, and octonions.
The third of three volumes on partial differential equations, this
is devoted to nonlinear PDE. It treats a number of equations of
classical continuum mechanics, including relativistic versions, as
well as various equations arising in differential geometry, such as
in the study of minimal surfaces, isometric imbedding, conformal
deformation, harmonic maps, and prescribed Gauss curvature. In
addition, some nonlinear diffusion problems are studied. It also
introduces such analytical tools as the theory of L^p Sobolev
spaces, Holder spaces, Hardy spaces, and Morrey spaces, and also a
development of Calderon-Zygmund theory and paradifferential
operator calculus. The book is targeted at graduate students in
mathematics and at professional mathematicians with an interest in
partial differential equations, mathematical physics, differential
geometry, harmonic analysis, and complex analysis. The third
edition further expands the material by incorporating new theorems
and applications throughout the book, and by deepening connections
and relating concepts across chapters. It includes new
sections on rigid body motion, on probabilistic results related to
random walks, on aspects of operator theory related to quantum
mechanics, on overdetermined systems, and on the Euler equation for
incompressible fluids. The appendices have also been updated
with additional results, ranging from weak convergence of measures
to the curvature of Kahler manifolds. Michael E. Taylor is a
Professor of Mathematics at the University of North Carolina,
Chapel Hill, NC. Review of first edition: “These volumes will be
read by several generations of readers eager to learn the modern
theory of partial differential equations of mathematical physics
and the analysis in which this theory is rooted.” (Peter Lax,
SIAM review, June 1998)
The first of three volumes on partial differential equations, this
one introduces basic examples arising in continuum mechanics,
electromagnetism, complex analysis and other areas, and develops a
number of tools for their solution, in particular Fourier analysis,
distribution theory, and Sobolev spaces. These tools are then
applied to the treatment of basic problems in linear PDE, including
the Laplace equation, heat equation, and wave equation, as well as
more general elliptic, parabolic, and hyperbolic equations. The
book is targeted at graduate students in mathematics and at
professional mathematicians with an interest in partial
differential equations, mathematical physics, differential
geometry, harmonic analysis, and complex analysis. The third
edition further expands the material by incorporating new theorems
and applications throughout the book, and by deepening connections
and relating concepts across chapters. In includes new
sections on rigid body motion, on probabilistic results related to
random walks, on aspects of operator theory related to quantum
mechanics, on overdetermined systems, and on the Euler equation for
incompressible fluids. The appendices have also been updated
with additional results, ranging from weak convergence of measures
to the curvature of Kahler manifolds.Michael E. Taylor is a
Professor of Mathematics at the University of North Carolina,
Chapel Hill, NC. Review of first edition: “These volumes will be
read by several generations of readers eager to learn the modern
theory of partial differential equations of mathematical physics
and the analysis in which this theory is rooted.” (Peter Lax,
SIAM review, June 1998)
This second in the series of three volumes builds upon the basic
theory of linear PDE given in volume 1, and pursues more advanced
topics. Analytical tools introduced here include pseudodifferential
operators, the functional analysis of self-adjoint operators, and
Wiener measure. The book also develops basic differential
geometrical concepts, centered about curvature. Topics covered
include spectral theory of elliptic differential operators, the
theory of scattering of waves by obstacles, index theory for Dirac
operators, and Brownian motion and diffusion. The book is targeted
at graduate students in mathematics and at professional
mathematicians with an interest in partial differential equations,
mathematical physics, differential geometry, harmonic analysis, and
complex analysis. The third edition further expands the material by
incorporating new theorems and applications throughout the book,
and by deepening connections and relating concepts across
chapters. It includes new sections on rigid body motion, on
probabilistic results related to random walks, on aspects of
operator theory related to quantum mechanics, on overdetermined
systems, and on the Euler equation for incompressible
fluids. The appendices have also been updated with
additional results, ranging from weak convergence of measures to
the curvature of Kahler manifolds. Michael E. Taylor is a Professor
of Mathematics at the University of North Carolina, Chapel Hill,
NC. Review of first edition: “These volumes will be read by
several generations of readers eager to learn the modern theory of
partial differential equations of mathematical physics and the
analysis in which this theory is rooted.” (Peter Lax, SIAM
review, June 1998)
This text introduces students to the theory and practice of
differential equations, which are fundamental to the mathematical
formulation of problems in physics, chemistry, biology, economics,
and other sciences. The book is ideally suited for undergraduate or
beginning graduate students in mathematics, and will also be useful
for students in the physical sciences and engineering who have
already taken a three-course calculus sequence. This second edition
incorporates much new material, including sections on the Laplace
transform and the matrix Laplace transform, a section devoted to
Bessel's equation, and sections on applications of variational
methods to geodesics and to rigid body motion. There is also a more
complete treatment of the Runge-Kutta scheme, as well as numerous
additions and improvements to the original text. Students finishing
this book will be well prepared for advanced studies in dynamical
systems, mathematical physics, and partial differential equations.
This text was produced for the second part of a two-part sequence
on advanced calculus, whose aim is to provide a firm logical
foundation for analysis. The first part treats analysis in one
variable, and the text at hand treats analysis in several
variables. After a review of topics from one-variable analysis and
linear algebra, the text treats in succession multivariable
differential calculus, including systems of differential equations,
and multivariable integral calculus. It builds on this to develop
calculus on surfaces in Euclidean space and also on manifolds. It
introduces differential forms and establishes a general Stokes
formula. It describes various applications of Stokes formula, from
harmonic functions to degree theory. The text then studies the
differential geometry of surfaces, including geodesics and
curvature, and makes contact with degree theory, via the
Gauss-Bonnet theorem. The text also takes up Fourier analysis, and
bridges this with results on surfaces, via Fourier analysis on
spheres and on compact matrix groups.
The mathematical formulations of problems in physics, economics,
biology, and other sciences are usually embodied in differential
equations. The analysis of the resulting equations then provides
new insight into the original problems. This book describes the
tools for performing that analysis. The first chapter treats single
differential equations, emphasizing linear and nonlinear first
order equations, linear second order equations, and a class of
nonlinear second order equations arising from Newton's laws. The
first order linear theory starts with a self-contained presentation
of the exponential and trigonometric functions, which plays a
central role in the subsequent development of this chapter. Chapter
2 provides a mini-course on linear algebra, giving detailed
treatments of linear transformations, determinants and
invertibility, eigenvalues and eigenvectors, and generalized
eigenvectors. This treatment is more detailed than that in most
differential equations texts, and provides a solid foundation for
the next two chapters. Chapter 3 studies linear systems of
differential equations. It starts with the matrix exponential,
melding material from Chapters 1 and 2, and uses this exponential
as a key tool in the linear theory. Chapter 4 deals with nonlinear
systems of differential equations. This uses all the material
developed in the first three chapters and moves it to a deeper
level. The chapter includes theoretical studies, such as the
fundamental existence and uniqueness theorem, but also has numerous
examples, arising from Newtonian physics, mathematical biology,
electrical circuits, and geometrical problems. These studies bring
in variational methods, a fertile source of nonlinear systems of
differential equations. The reader who works through this book will
be well prepared for advanced studies in dynamical systems,
mathematical physics, and partial differential equations.
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