Potential Theory

  • Lester L. Helms

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Lester L. Helms
    Pages 1-8
  3. Lester L. Helms
    Pages 9-62
  4. Lester L. Helms
    Pages 63-120
  5. Lester L. Helms
    Pages 121-164
  6. Lester L. Helms
    Pages 165-213
  7. Lester L. Helms
    Pages 215-258
  8. Lester L. Helms
    Pages 259-284
  9. Lester L. Helms
    Pages 285-320
  10. Lester L. Helms
    Pages 321-351
  11. Lester L. Helms
    Pages 353-390
  12. Lester L. Helms
    Pages 391-411
  13. Lester L. Helms
    Pages 413-451
  14. Lester L. Helms
    Pages 453-477
  15. Back Matter
    Pages 479-485

About this book

Introduction

Potential Theory presents a clear path from calculus to classical potential theory and beyond, with the aim of moving the reader into the area of mathematical research as quickly as possible. The subject matter is developed from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem, the author develops methods for constructing solutions of Laplace's equation on a region with prescribed values on the boundary of the region.

The latter half of the book addresses more advanced material aimed at those with the background of a senior undergraduate or beginning graduate course in real analysis. Starting with solutions of the Dirichlet problem subject to mixed boundary conditions on the simplest of regions, methods of morphing such solutions onto solutions of Poisson's equation on more general regions are developed using diffeomorphisms and the Perron-Wiener-Brelot method, culminating in application to Brownian motion.

In this new edition, many exercises have been added to reconnect the subject matter to the physical sciences. This book will undoubtedly be useful to graduate students and researchers in mathematics, physics, and engineering.

Keywords

Absorbing Boundary Barrier Brelot Brownian Motion Caauchy Initial Value Problem Capacity Cartan's Energy Principle Choquet Dirichlet Problem Fine Topology Gauss' Integral Green Function Greenian Set Harmonic Function Harmonic Measure Irregular Boundary Point Kelvin Tranformation Method of Images Neumann Problem Newtonian Potential Perron-Wiener-Brelot Method Poisson Integral Formula Poisson's Equation Reflecting Boundary Reflection Principle Regular Boundary Point Subnewtonian Kernel Wiener's Test Zaremba Cone Condition

Authors and affiliations

  • Lester L. Helms
    • 1
  1. 1.University of IllinoisUrbanaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4471-6422-7
  • Copyright Information Springer-Verlag London 2014
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4471-6421-0
  • Online ISBN 978-1-4471-6422-7
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • About this book