Potential Theory

  • Lester L. Helms

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Lester L. Helms
    Pages 1-6
  3. Lester L. Helms
    Pages 7-52
  4. Lester L. Helms
    Pages 53-105
  5. Lester L. Helms
    Pages 107-147
  6. Lester L. Helms
    Pages 149-196
  7. Lester L. Helms
    Pages 197-240
  8. Lester L. Helms
    Pages 241-265
  9. Lester L. Helms
    Pages 267-301
  10. Lester L. Helms
    Pages 303-331
  11. Lester L. Helms
    Pages 333-369
  12. Lester L. Helms
    Pages 371-389
  13. Lester L. Helms
    Pages 391-429
  14. Back Matter
    Pages 431-441

About this book

Introduction

Aimed at graduate students and researchers in mathematics, physics, and engineering, this book presents a clear path from calculus to classical potential theory and beyond, moving the reader into a fertile area of mathematical research as quickly as possible. The author revises and updates material from his classic work, Introduction to Potential Theory (1969), to provide a modern text that introduces all the important concepts of classical potential theory.

In the first half of the book, the subject matter is developed meticulously from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem of the calculus, the author develops methods for constructing solutions of Laplace’s equation on a region with prescribed values on the boundary of the region.

The second half addresses more advanced material aimed at those with a background of a senior undergraduate or beginning graduate course in real analysis. For specialized regions, namely spherical chips, solutions of Laplace’s equation are constructed having prescribed normal derivatives on the flat portion of the boundary and prescribed values on the remaining portion of the boundary. By means of transformations known as diffeomorphisms, these solutions are morphed into local solutions on regions with curved boundaries. The Perron-Weiner-Brelot method is then used to construct global solutions for elliptic partial differential equations involving a mixture of prescribed values of a boundary differential operator on part of the boundary and prescribed values on the remainder of the boundary.

Keywords

Boundary Dirichlet Elliptic Oblique Potential Potential theory mathematics partial differential equation physics

Editors and affiliations

  • Lester L. Helms
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-84882-319-8
  • Copyright Information Springer-Verlag London 2009
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-84882-318-1
  • Online ISBN 978-1-84882-319-8
  • Series Print ISSN 0172-5939
  • About this book