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Author Bland, D. R. author Solutions of Laplace's Equation [electronic resource] / by D. R. Bland Dordrecht : Springer Netherlands, 1961 http://dx.doi.org/10.1007/978-94-011-7694-1 VIII, 98 p. online resource

SUMMARY

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 equaยญ 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 physiยญ cal sciences; special techniques applicable only to the soluยญ 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

CONTENT

1. Occurrence and Derivation of Laplace's Equation -- 1. Situations in which Laplace's equation arises -- 2. Laplace's equation in orthogonal curvilinear co-ordinates -- 3. Laplace's equation in particular co-ordinate systems -- 2. The Method of Separation of Variables -- 1. Rectangular Cartesian co-ordinates -- 2. Temperature distribution in a rectangular metal block -- 3. The analogous electrostatic problem -- 4. Cylindrical polar co-ordinates -- 5. Spherical polar co-ordinates -- 3. Bessel Functions -- 1. An infinite series solution of Bessel's equation -- 2. Bessel functions of the second kind -- 3. Derivatives of Bessel functions and recurrence formulae -- 4. Modified Bessel functions -- 5. Behaviour of Bessel functions at zero and infinity -- 6. Series of zero order Bessel functions -- 4. Solutions using Cylindrical Polar Co-ordinates -- 1. Form of solutions of Laplace's equation -- 2. An infinite cylinder in a uniform field -- 3. A particular solid of revolution in a uniform field -- 4. Axi-symmetric temperature distributions in a cylinder -- 5. Legendre Polynomials -- 1. Solution in series of Legendre's equation -- 2. Associated Legendre functions -- 3. Derivatives and recurrence formulae for Legendre polynomials -- 4. Series of Legendre polynomials -- 6. Solutions using Spherical Polar Co-ordinates -- 1. Form of solutions of Laplace's equation -- 2. Sphere moving in a liquid at rest at infinity -- 3. A charged conducting sphere in a uniform electric field -- 4. Dielectric sphere in a uniform electric field -- 5. Axi-symmetric temperature distributions in a hollow sphere -- 6. Flow past a nearly spherical body -- 7. Sources, sinks and doublets -- 8. Doublet in a fluid bounded by a sphere -- 9. Doublet in a cavity in a dielectric medium

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