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The 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.
Front Matter
Occurrence and Derivation of Laplace’s Equation
The Method of Separation of Variables
Bessel Functions
Solutions Using Cylindrical Polar Co-ordinates
Legendre Polynomials
Solutions Using Spherical Polar Co-ordinates
Back Matter