This books results from lectures given by the author at the Rijks Universiteit at Utrecht. These lectures were part of a series on history of contemporary mathematics in which the author of the present text and his collegues treated several topics in the history of present-day mathematics.
Originally the title of these lectures was: ‘Gauss, Riemann and Dirichlet’s principle’ but writing this text in the form of a book it appeared that this subject leads to a broader domain. The principle of Dirichlet has played an important role in the development of mathematical analysis; there are connections with complex function theory, the calculus of variations, functional analysis. For a good understanding it appeared to be necessary to incorporate some passages in modern mathematical form. They are presented in such a way that the reader is hoped to be able to distinghuish them from the historical de- velopments. There are also passages which are somewhat further away from the central theme — i.e. Dirichlet’s principle — than potential theory and complex function theory but it is believed that they contribute to a better understanding of the development of mathematics in an important period. Attention is given to the physical backgrounds of the subject.
The present book is therefore a mixture of history and mathematics. There seem to be few books written in this style but the author believes that they are useful in a time when the interest in the history of contemporary mathematics is growing. The author hopes that any mathe- matician can profit from this book in view of his mathematical erudition. Profound knowledge of the subject is not required; any student with a certain amount of mathematical erudition will be able to read the book. An extensive bibliography will be helpful for those who want to read themselves in the sources.
I wish to express my thanks to E.M.J.Bertin, H.J.M.Bos and T.A Springer for their most valuable critical remarks. I thank Miss W. Jenner for the accurate way in which she typed the manuscript.
AUGUST 1975 A. F. MONNA
Author(s): A. F. Monna
Publisher: Oosthoek, Scheltema & Holkema
Year: 1975
Language: English
Commentary: No attempt at file size reduction
Tags: Potential Theory; Dirichlet's Principle; Calculus of Variations
Foreword v
Introduction 1
I The origins of potential theory 5
II Variational principles 11
1 C.F. Gauss 17
2 W.Thomson (Lord Kelvin) 25
3 P.G.Lejeune-Dirichlet 27
4 B.Riemann 32
III Criticism and the developments in post critical years 35
1 Criticism 35
2 Developments in the post critical years 43
3 Theory of maxima and minima 50
IV Hilbert and the calculus of variations 55
1 The direct method 55
2 Hilbert’s problems 60
V Complex functions and potentials 67
1 Riemann and complex function theory 68
2 The influence of physics 82
3 K. Weierstrass and analytic functions 98
VI Functional-analytic approach 107
VII Modern developments 119
1 The final version of Dirichlet’s principle 179
2 Axiomatics 124
References 129
Index 137
