Partial Differential Equations: Basic Theory (Texts in Applied Mathematics) [Paperback]

Michael E. Taylor (Author)

Book Description

ISBN-10: 0387946543 | ISBN-13: 978-0387946542 | Publication Date: June 25, 1996 | Edition: Corrected

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.

Editorial Reviews

Review

“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.” (SIAM Review, June 1998) --This text refers to the Hardcover edition.

From the Back Cover

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. In this second edition, there are seven new sections including Sobolev spaces on rough domains, boundary layer phenomena for the heat equation, the space of pseudodifferential operators of harmonic oscillator type, and an index formula for elliptic systems of such operators. In addition, several other sections have been substantially rewritten, and numerous others polished to reflect insights obtained through the use of these books over time. 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.” (SIAM Review, June 1998) --This text refers to the Hardcover edition.

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Product Details

• Paperback: 604 pages
• Publisher: Springer; Corrected edition (June 25, 1996)
• Language: English
• ISBN-10: 0387946543
• ISBN-13: 978-0387946542
• Product Dimensions: 9.2 x 6 x 1.2 inches
• Shipping Weight: 1.8 pounds (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #1,520,746 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

14 of 14 people found the following review helpful:

4.0 out of 5 stars Great PDE book, September 22, 2006

By 

Ian Langmore (New York, NY, USA) - See all my reviews
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This review is from: Partial Differential Equations I: Basic Theory (Applied Mathematical Sciences) (Hardcover)

I like this book. Right from the beginning, Taylor uses concepts from smooth manifold theory to obtain results in a coordinate free fashion (on a Riemannian/Lorenz manifold) when possible. At the same time, he will go to coordinates when necessary, or even revert to R^n when stronger results are possible there. As an added "bonus" the book contains surprisingly few errors, or at least I found less than usual, so maybe I'm getting slow!

I didn't find this to be an introductory (as in first year of grad school) level book. The proofs are often given in less detail than those found in the standard introductory graduate texts. Reading and understanding this text requires knowledge of Real Analysis and smooth manifolds/differential geometry at the introductory graduate level. Fourier transforms and distributions are built up from scratch, but the treatment is quick and probably assumes you have some previous knowledge. Note that my opinion is formed after using this book for self-study, so you can probably relax the prerequisites slightly if this book is used as part of a course.

Another point worth mentioning is that the exercises are often quite easier than verifying the individual proofs in the chapter.

Why not 5 stars? I found the book and chapter introductions to be quite uninspiring and not very useful. He essentially lists off the material that will be covered with no thought to motivation. I like to read something about why one particular approach is necessary or better or just different than another approach. Simply listing off concepts to be covered has little meaning if one has no idea what those concepts mean! The discussion of results is also rather sparse, sort of like Rudin.

12 of 15 people found the following review helpful:

5.0 out of 5 stars Complete, accurate and well-written., May 10, 2000

By 

Bernardo Vargas - See all my reviews

This review is from: Partial Differential Equations I: Basic Theory (Applied Mathematical Sciences) (Hardcover)

The author accomplished the goal of presenting this broad and many-faceted subject in a thorough and comprehensive manner. Beginning from the fundamentals of ODE theory to the most sophisticated methods for solving important PDE's of mathematical physics, this series of three volumes comprises all what a modern analyst must know about the topic and much more.

The contents of volume 1 are: Basic theory of ODE; Laplace and wave equations; Fourier analysis, distributions, and constant-coefficient linear PDE; Sobolev spaces; linear elliptic equations; linear evolution equations. Appendices: Outline of functional analysis; manifolds, vector bundles, and Lie groups.

Originally intended for graduate students and working mathematicians, most of the material is suitable for advanced undergraduate courses. Includes excercises for each section and extensive references.