Dirichlet's principle: A mathematical comedy of errors and its influence on the development of analysis

152 Pages·1975·29.431 MB·English

by  A. F. Monna

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THOMASJ . BATA LIBRARY
TRENT UNIVERSITY
Dirichlet’s principle

Dirichlet’s principle
a mathematical comedy of errors
and its influence on the development
of analysis
DR. A. F. MONNA
Professor of Mathematics
Rijksuniversiteit Utrecht
1975
Oosthoek, Scheltema & Holkema
Utrecht, The Netherlands
© 1975 Scheltema & Holkema by
No part of this book may be reproduced in any form by print, photo-
print, microfilm or any other means without written permission from
the publisher.
ISBN 90 313 O175 2
The comedy of errors
All’s well that ends well
Shakespeare
Foreword
This books results from lectures given by the author at the Rijks Uni-
versiteit 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 mathe-
matics.
Originally the title of these lectures was: ‘Gauss, Riemann and Di-
richlet’s principle’ but writing this text in the form of a book it ap-
peared that this subject leads to a broader domain. The principle of
Dirichlet has played an important role in the development of mathe-
matical analysis; there are connections with complex function theory,
the calculus of variations, functional analysis. For a good under-
standing 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 con-
tribute 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 con-
temporary 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
Vv
RIGDEL
FOREWORD
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
VI
Contents
Foreword v
Introduction 1
1 The origins of potential theory 5
ut Variational principles 11
1 C.F.Gauss 17
2 W.Thomson (Lord Kelvin) 25
3 P.G.Lejeune-Dirichlet 27
4 B.Riemann 32
ut 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
vit Modern developments 119
1 The final version of Dirichlet’s principle 179
2 Axiomatics 124
References 129
Index 137
vu