Customer Reviews

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

5 star:		(1)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

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.

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.