# Solutions of Laplace’s Equation

• Authors
• D. R. Bland
Book

Part of the Library of Mathematics book series (LIMA)

1. Front Matter
Pages N2-viii
2. D. R. Bland
Pages 1-14
3. D. R. Bland
Pages 15-28
4. D. R. Bland
Pages 29-47
5. D. R. Bland
Pages 48-60
6. D. R. Bland
Pages 61-77
7. D. R. Bland
Pages 78-96
8. Back Matter
Pages 97-98

### Introduction

THIS book is an introduction both to Laplace's equation and its solutions and to a general method of treating partial differential equations. Chapter 1 discusses vector fields and shows how Laplace's equation arises for steady fields which are irrotational and solenoidal. In the second chapter the method of separation of variables is introduced and used to reduce each partial differential equation, Laplace's equa­ tion in different co-ordinate systems, to three ordinary differential equations. Chapters 3 and 5 are concerned with the solutions of two of these ordinary differential equations, which lead to treatments of Bessel functions and Legendre polynomials. Chapters 4 and 6 show how such solutions are combined to solve particular problems. This general method of approach has been adopted because it can be applied to other scalar and vector fields arising in the physi­ cal sciences; special techniques applicable only to the solu­ tions of Laplace's equation have been omitted. In particular generating functions have been relegated to exercises. After mastering the content of this book, the reader will have methods at his disposal to enable him to look for solutions of other partial differential equations. The author would like to thank Dr. W. Ledermann for his criticism of the first draft of this book. D. R. BLAND The University, Sussex. v Contents Preface page v 1. Occurrence and Derivation of Laplace's Equation 1. Situations in which Laplace's equation arises 1 2. Laplace's equation in orthogonal curvilinear co-ordinates 8 3.

### Keywords

Finite behavior derivation derivative distribution equation evolution field form function functions variable

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-011-7694-1