MATHEMATISCHEN WISSENSCHAFTEN

DIE GRUNDLEHREN DER

MATHEMATISCHEN WISSENSCHAFTEN IN EINZELDARSTELLUNGEN MIT BESONDERER

BEROCKSICHTIGUNG DER ANWENDUNGSGEBIETE

GEMEINSAM MIT

W. BLASCHKE M. BORN HAMBURG G{}TTINGEN

HERAUSGEGEBEN VON

R. COURANT G{}TTINGEN

BAND XXXI

FOUNDATIONS OF POTENTIAL THEORY

BY

OLIVER DIMON KELLOGG

Co RUNGE G{}TTINGEN

SPRINGER-VERLAG

BERLIN· HEIDELBERG· NEW YORK

1967

FOUNDATIONS OF

POTENTIAL THEORY BY

OLIVER DIMON KELLOGG PROFESSOR OF MATHEMATICS IN HARVARD UNIVERSITY

CAMBRIDGE· MASSACHUSETTS· U.S.A.

WITH 30 FIGURES

REPRINT FROM THE FIRST EDITION OF 1929

SPRINGER-VERLAG

BERLIN· HEIDELBERG· NEW YORK

1967

Aile Rechte, insbesondere das der Ubersetzung in fremde Sprachen, vorbehalten. Ohne ausdriick­liehe Genehmigung des Verlages ist es auch nicht gestattet, dieses Buch oder Teile daraus auf

photomechanischem Wege (Photokopie, Mikrokopie) oder auf andere Art zu vervielfaltigen

Library of Congress Catalog Card Number 67-29862

Titel Nr. \014

Softcover reprint of the hardcover 1st edition 1967

ISBN-13: 978-3-642-86750-7 e-ISBN-I3: 978-3-642-86748-4 001: 10.1007/978-3-642-86748-4

Preface. The present volume gives a systematic treatment of potential

functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose: first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals; and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to the periodical literature of the day.

It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, in order that the book may present sound ideals to the student, and also serve the mathematician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem (Gauss', or Green's theorem) on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to ·the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Dirichlet problem.

Exercises are introduced in the conviction that no mastery of a mathematical subject is possible without working with it. They are designed primarily to illustrate or extend the theory, although the desirability of requiring an occasional concrete numerical result has not been lost sight of.

Grateful acknowledgements are due to numerous friends on both sides of the Atlantic for their kind interest in the work. It is to my colleague Professor COOLIDGE that lowe the first suggestion to under­take it. To Professor OSGOOD I am indebted for constant encouragement and wise counsel at many points. For a careful reading of the manuscript and for helpful comment, I am grateful to Dr. ALEXANDER WEINSTEIN, of Breslau; and for substantial help with the proof, I wish to thank my pupil Mr. F. E. ULRICH .. It is also a pleasure to acknowledge the generous attitude, the unfailing courtesy, and the ready cooperation of the publisher.

Cambridge, Mass. August, 1929.

O. D. Kellogg.

Contents.

Chapter I. The Force of Gravity.

1. The Subject Matter of Potential Theory . . . . . . . . . . . .. 1 2. Newton's Law • . . . . . . . . . . . . . . . . . . . . . . .. 2 3. Interpretation of Newton's Law for Continuously Distributed Bodies 3 4. Forces Due to Special Bodies 4 5. Material Curves, or Wires 8 6. Material Surfaces or Laminas 10 7. Curved Laminas • . . . . . 12 8. Ordinary Bodies, or Volume Distributions 15 9. The Force at Points of the Attracting Masses 17

10. Legitimacy of the Amplified Statement of Newton's Law; Attraction between Bodies . . . . . • . . . . . . . . . . . 22

II. Presence of the Couple; Centrobaric Bodies; Specific Force. . . . . . 26

Chapter II.

Fields of Force. 1. Fields of Force and Other Vector Fields . 2. Lines of Force. . . . . . . . . . . 3. Velocity Fields ......... . 4. Expansion, or Divergence of a Field. 5. The Divergence Theorem .... 6. Flux of Force; Solenoidal Fields 7. Gauss' Integral 8. Sources and Sinks . . . 9. General Flows of Fluids; Equation of Continuity .

Chapter III. The Potential.

1. Work and Potential Energy 2. Equipotential Surfaces . . . . . . . . . . 3. Potentials of Special Distributions. . . . . 4. The Potential of a Homogeneous Circumf~rence 5. Two Dimensional Problems; The Logarithmic Potential 6. Magnetic Particles . . . . . . . . . . . 7. Magnetic Shells, or Double Distributions. 8. Irrotational Flow 9. Stokes' Theorem . . . . . .

10. Flow of Heat . . . . . . . II. The Energy of Distributions 12. Reciprocity; Gauss' Theorem of the Arithmetic Mean

28 28 31 34 37 40 42 44 45

48 54 55 58 62 65 66 69 72 76 79 82

Contents. VII

Chapter IV.

The Divergence Theorem. 1. Purpose of the Chapter .......... 84 2. The Divergence Theorem for Normal Regions. 85 3. First Extension Principle 88 4. Stokes' Theorem . . . . . 89 5. Sets of Points . . . . . . 91 6. The Heine-Borel Theorem 94 7. Functions of One Variable; Regular Curves 97 8. Functions of Two Variables; Regular Surfaces 100 9. Functions of Three Variables . . . . . . . . 113

10. Second Extension Principle; The Divergence Theorem for Regular Re-gions . . . . . . . . . . . . . . . . . . . . . . . . . 113

11. Lightening of the Requirements with Respect to the Field. 119 12. Stokes' Theorem for Regular Surfaces . . . . . . . . . . 121

Chapter V.

Properties of Newtonian Potentials at Points of Free Space. 1. Derivatives; Laplace's Equation .• 2. Development of Potentials in Series . . . . 3. Legendre Polynomials ......... . 4. Analytic Character of Newtonian Potentials. 5. Spherical Harmonics . . . . . . . . . . . 6. Development in Series of Spherical Harmonics 7. Development Valid at Great Distances. . . . 8. Behavior of Newtonian Potentials at Great Distances

Chapter VI.

Properties of Newtonian Potentials at Points Occupied by Masses.

121 124 125 135 139 141 143 144

1. Character of the Problem . . . . . . . 146 2. Lemmas on Improper Integrals . . . . 146 3. The Potentials of Volume Distributions 150 4. Lemmas on Surfaces . . . . • . . . . 157 5. The Potentials of Surface Distributions. 160 6. The Potentials of Double Distributions 166 7. The Discontinuities of Logarithmic Potentials 172

Chapter VII. Potentials as Solutions of Laplace's Equation; Electrostatics.

1. Electrostatics in Homogeneous Media . . . . . . . 175 2. The Electrostatic Problem for a Spherical Conductor 176 3. General Coordinates . • • . . . . . . . 178 4. Ellipsoidal Coordinates . . . . . . . . . . . . . 184 5. The Conductor Problem for the Ellipsoid. . . . . 188 6. The Potential of the Solid Homogeneous Ellipsoid 192 7. Remarks on the Analytic Continuation of Potentials 196 8. Further Examples Leading to Solutions of Laplace's Equation. 198 9. Electrostatics; Non-homogeneous Media •......•.. 206

Chapter VIII. Harmonic Functions.

1. Theorems of Uniqueness . . . . . . . . . . . . . . . . . . 211 2. Relations on the Boundary between Pairs of Harmonic Functions 215

vrn Contents.

3. Infinite Regions . . . . . . . . . . . . . . . . 4. Any Harmonic Function is a Newtonian Potential 5. Uniqueness of Distributions Producing a Potential 6. Further Consequences of Green's Third Identity 7. The Converse of Gauss' Theorem ....

Chapter IX.

Electric Images i Green's Function.

I. Electric Images . . . . . . . . . 2. Inversion; Kelvin Transformations ........ . 3. Green's Function ............... . 4. Poisson's Integral; Existence Theorem for the Sphere 5. Other Existence Theorems . . . . . . . . . . . .

Chapter X.

Sequences of Harmonic Functions.

I. Harnack's First Theorem on Convergence. 2. Expansions in Spherical Harmonics . . . 3. Series of Zonal Harmonics . . . . . . . 4. Convergence on the Surface of the Sphere 5. The Continuation of Harmonic Functions. 6. Harnack's Inequality and Second Convergence Theorem 7. Further Convergence Theorems . . . . . . 8. Isolated Singularities of Harmonic Functions 9. Equipotential Surfaces . . . . . . . . .

Chapter XI. Fundamental Existence Theorems.

216 218 220 223 224

228 231 236 240 244

248 251 254 256 259 262 264 268 273

I. Historical Introduction . . . . . . . . . . . . . . . . . . . . . . 277 2. Formulation of the Dirichlet and Neumann Problems in Terms of Inte-

gral Equations . . . . . . . . . . . . . . . . . . . . . . .. 286 3. Solution of Integral Equations for Small Values of the Parameter • 287 4. The Resolvent . . . . . . . . . . . . . . . . . . . . 289 5. The Quotient Form for the Resolvent . . . . . . . . . 290 6. Linear Dependence; Orthogonal and Biorthogonal Sets of Fu"ctions 292 7. The Homogeneous Integral Equations . . . . . . . . . . . . .. 294 8. The Non-homogeneous Integral Equation; Summary of Results for Con-

tinuous Kernels . . . . . . . . . . . . . . . . . . 297 9. Preliminary Study of the Kernel of Potential Theory 299

10. The Integral Equation with Discontinuous Kernel. 307 II. The Characteristic Numbers of the Special Kernel. . 309 12. Solution of the Boundary Value Problems . . . . . 311 13. Further Consideration of the Dirichlet Problem; Superharmonic and

Subharmonic Functions. . . . . . . . . . . . . . . . . . . . . . 315 14. Approximation to a Given Domain by the Domains of a Nested Sequence 317 15. The Construction of a Sequence Defining the Solution of the Dirichlet

Problem. . . . . . . . . . . . 322 16. Extensions; Further Properties of U . 323 17. Barriers. . . . . . . . . . 326 18. The Construction of Barriers 328 19. Capacity 330 20. Exceptional Points. . . . . 334

Contents.

Chapter XII. The Logarithmic Potential.

I. The Relation of Logarithmic to Newtonian Potentials. 2. Analytic Functions of a Complex Variable •..... 3. The Cauchy-Riemann Differential Equations . . . • . 4. Geometric Significance of the Existence of the Derivative. 5. Cauchy's Integral Theorem . . . . . . 6. Cauchy's Integral. . . . . . . . . . . 7. The Continuation of Analytic Functions 8. Developments in Fourier Series • . 9. The Convergence of Fourier Series

10. Conformal Mapping ....... . II. Green's Function for Regions of the Plane. 12. Green's Function and Conformal Mapping 13. The Mapping of Polygons

Bibliographical Notes.

Index

IX

338 340 341 343 344 348 351 353 355 359 363 365 370

377 379