Customer Reviews

Linear differential operators.

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A very intuitive (geometrical) exposition of matrix calculus, adjoint problems, bilinear identity and Green's function (and more). If you really want to understand these concepts, read this masterpiece!

As the other reviewers have said, this is a master piece for various reasons. Lanczos is famous for his work on linear operators (and efficient algorithms to find a subset of eigenvalues). Moreover, he has an "atomistic" (his words) view of differential equations, very close to the founding father's one (Euler, Lagrange,...).

A modern book on linear operators begins with the abstract concept of function space as a vector space, of scalar product as integrals,... The approach is powerful but somehow we loose our good intuition about differential operators.

Lanczos begins with the simplest of differential equations and use a discretization scheme (very natural to anybody who has used a computer to solve differential equations) to show how a differential equation transforms into a system a linear algebraic equation. It is then obvious that this system is undetermined and has to be supplemented by enough boundary condition to be solvable. From here, during the third chapters, Lanczos develops the concept of linear systems and general (n x m) matrices, the case of over and under determination, the compatibility conditions, ...
It is only after these discussions that he returns (chapter 4) to the function space and develops the operator approach and the role of boundary conditions in over and under-determination of solutions and the place of the adjoint operators. The remaining of the book develops these concepts : chp5 is devoted to Green's function and hermitian problems, chap7 to Sturm-Liouville,... The last chapter is devoted to numerical techniques, amazing if one think that the book was written at the very beginning of computers, which is a gem by itself.

Some mathematics and physics writers stand head and shoulders above the rest. Goldstein...Liboff...Morrison...Morse and Feshbach...and Lanczos. A joy to read, if you are both mathematically and verbally inclined.

This book has material I've found in no other book. Lanczos is a pleasure to read -- his writing is clear, elegant, and entertainingly opinionated. I've liked every book of his that I've read.

Somebody writen:

"Some mathematics and physics writers stand head and shoulders above the rest. Goldstein...Liboff...Morrison...Morse and Feshbach...and Lanczos. A joy to read, if you are both mathematically and verbally inclined."

I think some mathematics and physics writers stand head and shoulders above even Goldstein...Liboff...Morrison...Morse and Feshbach. It is the case of Lanczos and Dirac.

Lanczos is a master of insightful presentation and, I think, his ability reached its climax in this book. Pure mathematicians may object to the presentation given in this book and even to its title as "Linear Operators". But who minds? He already declares that "the present book is written from the applied angle.." (Section 4.2, page 165).

I had run into this book more than 35 years ago, when I was a graduate research assistant in applied mathematics. I was literally struggling with the technicalities of Lebesgue's Integration Theory and Functional Analysis, through the books of authors like McShane, Dunford &Schwartz . It had immediately captivated me as if somebody had unexpectedly pulled me out of a bog into the light. The essentials of the realm of function spaces, differential and integral operators, Green functions and integral transformations had appeared to a young mind almost in a flash after a painless and self-flowing reading. Today a professional engineer in his sixty, I steel remember that emotion and feel the same captivation when I collect this book from my Library and page through it. It is my sincere belief, and wish at the same time, that no young graduate of mathematics,physics and even engineering should miss the chance of reading this masterpiece.