# INTRODUCTION TO APPLIED MATHEMATICS

## GILBERT STRANG

## Preface

I believe that the teaching of applied mathematics needs a fresh approach. That opinion seems to be widely shared. Many of the textbooks in use today were written a generation ago, and they cannot reflect the ideas (or the algorithms) that have brought such significant change. Certainly there are things that will never be different-but even for the solution of Laplace's equation a great deal is new. -In addition to Fourier series and complex variables, it is time to see fast transforms and finite elements. Topics like stability and optimization and matrix methods have earned a more central place-perhaps at the expense of series solutions of differential equations.

Applied mathematics is alive and very vigorous. That ought to be reflected in our teaching. In my own class I became convinced that the textbook is crucial. It must provide a framework into which the applications will fit. A good course has a clear purpose, and you can sense that it is there. It is a pleasure to teach a subject when it is moving forward, and this one is-but the book has to share in that spirit and help to establish it.

The central topics are differential equations and matrix equations-the continuous and discrete. They reinforce each other because they go in parallel. On one side is calculus; on the other is algebra. Differential equations and their transforms are classical, but still beautiful and essential. Nothing is out of date about Fourier! At the same time this course must develop the discrete analogies, in which potential differences rather than derivatives drive the flow. Those analogies are not difficult; they are basic, and I have found that they are welcome. To see the cooperation between calculus and linear algebra is to see one of the best parts of modern applied mathematics.

I am also convinced that these equations are better understood-they are more concrete and useful-when we present at the same time the algorithms that solve them. Those algorithms give support to the theory. And in general, numerical methods belong with the problems they solve. I do not think that the Fast Fourier Transform and difference equations and numerical linear algebra belong only in some uncertain future course. It is *this course* that should recognize what the computer can do, without being dominated by it.

Perhaps I may give special mention to the role of linear algebra. If any subject has become indispensable, that is it. The days of explaining matrix notation in an appendix are gone; this book assumes a working knowledge of matrices. It begins with the solution of Ax = b - by elimination and its factorization of A into LU or LDL^{T}, not by computing A^{-1}. A fuller treatment is given in my textbook *Linear Algebra and Its Applications* (Academic Press). Here we start with the essential facts and move on rapidly to put them to use.

I do not think that the right approach is to model a few isolated examples. The important goal is to find ideas that are shared by a wide range of applications. This is the contribution a book can make, to recognize and explain the underlying pattern. We need to present Kirchhoff's laws, and to see how they lead in the continuous case to the curl and divergence, but we hope to avoid a total and fatal immersion into vector calculus-which has too frequently replaced applied mathematics, and taken all the fun out of it.

It seems natural for the discrete case to come first, but Chapter I *should not be too slow.* It is Chapter 2 that points out the triple product *A ^{T}CA* in the equations of equilibrium, and Chapter 3 finds that framework repeated by differential equations. Applications and examples are given throughout. Where the theory strengthens the understanding, it is provided-but this is not a book about proofs.

After the equations are formulated they need to be solved. Chapter 4 uses Fourier methods and complex variables, Chapter 5 uses numerical methods. It presents scientific computing as an integral part of applied mathematics, with the same successes (especially through Fourier) and the same difficulties (including stability in Chapter 6). Orthogonality is fundamental, as it is for analytical methods.

The book is a text for applied mathematics and advanced calculus and engineering mathematics. It aims to explain what is essential, as far as possible within one book and one year. It also reaches beyond the usual courses, to introduce more recent ideas:

2.5 The Kalman filter

5.3 Iterative methods for Ax = b and Ax = ax

5.4 Finite elements

6.2 Chaos and strange attractors

6.6 Shock waves and solutions

7.1 Combinatorial optimization

7.4 Maximal flows and minimal cuts in networks

8.2 Karmarkar's method for linear programming

You will not have time for all of these; I do not. Those that are unfamiliar may stay that way; no objection. Nevertheless they are part of this subject and they fit perfectly into its framework - even Karmarkar's method has a place, whether or not it fulfills everything that has been promised. (It outdoes the simplex method on some problems but by no means on all.) It solves the piecewise linear equations of mathematical programming through a series of linear equations, whose coefficient matrices are again *A ^{T}CA*. Finite elements and recursive filters and Laplace's equation fit this same pattern. In my experience the framework is seen and understood by the reader, and appreciated.

The whole subject is extremely coherent, and sections or chapters which are left for reference are no less valuable on that account. In fact this is also a textbook on numerical analysis (with applications included in the course, not separated) and on optimization (emphasizing the quadratic problems of Chapter 2, the network flows of Chapter 7, and the duality theory of Chapter 8). The book starts with ordinary differential equations and makes the transition to partial differential equations; that is not a great obstacle. It presents the minimum principles of least energy and least action, which are more subtle than equations and sometimes more revealing. But the emphasis must go to equilibrium equations (*boundary-value problems*) and dynamic equations (*initial-value problems*). Those are the central questions, continuous and discrete.

Applied mathematics is a big subject, and this is not a short book. Its goal is to be as useful as possible to the reader, in class and out. Engineering and scientific applications play a larger part than before; infinite series play a smaller part. Those series need to give way, in teaching as they have done in practice, to a more direct approach to the solution. We intend to try-working always with specific ,examples-to combine the algorithms with the theory. That is the way to work and I believe it is also the way to learn. The effort to teach what students will need and use is absolutely worthwhile.

A personal note before the book begins. These years of writing have been a tremendous pleasure, and one reason was the help offered by friends. It was shamelessly accepted, and what can I do in return? A public acknowledgment is made at the end, of my gratitude for their encouragement. It is what keeps an author going.

In one respect this is a special adventure. I decided that it must be possible to see the book all the way through. I care too much about the subject to mail in a manuscript and say goodbye. Therefore Wellesley-Cambridge Press was created, in order to do it right. The book was printed in the normal way, and it should be handled more efficiently than usual. Bookstores (and individual readers) will need a new address and telephone number, given below. No representative will call! I have to depend on those who enjoy it to say so. More than ever it must stand on its own - if it can. I hope you like the book.

Gilbert Strang Wellesley-Cambridge Press M.I.T. Box 157 Room 2-240 Wellesley MA 02181 Cambridge MA 02139 (617)235-9537 (617) 253@4383

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