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Introduction to fundamentals of potential functions covers: the force of gravity, fields of force, potentials, harmonic functions, electric images and Green’s function, sequences of harmonic functions, fundamental existence theorems, the logarithmic potential and much more. Detailed proofs rigorously worked out. 1929 ed.
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Beautiful mathematics
A beautiful presentation which I much enjoyed in grad school.Now I need to return to the subject, but I can't find it in my lib.So I'm ordering another copy and looking forward to reading it again!

THE book on classical potential theory
There are few books left in print on this captivating subject from theoretical mechanics. Kellogg's presentation is clear and mathematically formal. This is an excellent reprint from Dover.

Now, if Dover would onlybring back into print MacMillan's "The Theory of the Potential"and maybe try to reprint Sternberg & Smith's U. of Toronto Presspublished (and out of print) "The Theory of the Potential andSpherical Harmonics", they would have themselves a trilogy to cornerthe science book market for the classical treatment of potentialtheory.

Kellogg stands fairly well on its own, but the other two booksjust mentioned fill-in Kellogg's gaps. ... Read more

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From the reviews of the second edition: "The new methods of complex manifold theory are very useful tools for investigations in algebraic geometry, complex function theory, differential operators and so on. The differential geometrical methods of this theory were developed essentially under the influence of Professor S.-S. Chern's works. The present book is a second edition... It can serve as an introduction to, and a survey of, this theory and is based on the author's lectures held at the University of California and at a summer seminar of the Canadian Mathematical Congress.... The text is illustrated by many examples... The book is warmly recommended to everyone interested in complex differential geometry." #Acta Scientiarum Mathematicarum, 41, 3-4# ... Read more

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This book bridges the gap between the classic texts on potential theory and modern books on applied geophysics. Introductory chapters discuss potential theory with emphasis on those aspects particularly important to earth scientists, such as Laplace's equation, Newtonian potential, magnetic and electrostatic fields, conduction of heat, and spherical harmonic analysis. Later chapters apply these theoretical concepts specifically to the interpretation of gravity and magnetic anomalies, with emphasis on anomalies caused by crustal and lithospheric sources. The book is ideal for graduate students and researchers in applied geophysics. ... Read more

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Outstanding
This is a surprisingly well explained and self contained book of the subject. It covers the whole range of applications and theory that a geophysicist will need, going from the whole earth to regional and local applications.

Remarkable points are:
The mathematical foundation of the potential
Spherical harmonics in gravity and magnetic fields
Implementation and computation of local anomalies
Forward and inverse problems
Application of Fourier transform to potential fields.

This was the textbook for my Potential Fields class where we covered the first six chapters; I alsousedit for my Inverse Theory class as a referenceand I found in this title all I needed for my final project . All the persons that I recommend the book bought their own copy including my professor.

The book expands subjects that are barely touched (or considered a curiosity) by general books of Exploration Geophysics, like for example:

downward continuation
pseudo-gravity
annihilators
non-linear inverse problems, etc.

The exercises at the end of the chapters are challenging and complement the material very well, and the FORTRANscripts of the appendix are really helpful.

This title isdefinitely a must have for anyone involved in geophysical applications of Gravity and Magnetism in either exploration or basic research.

Required background: Vector calculus and differential equations. ... Read more

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Customer Reviews (1)

Almost a must.
This is a classic, read and cited by many. In describing monographs on stochastic processes, Jacod and Shiryaev (1987) call it``the very complete book", and Ethier and Kurtz (1986) - ``most recentpresentation". Actually, it is even difficult to find a serious bookon the subject that does not cite the volumes ofDellacherie and Meyer. Despite its age,this monograph is stillalmost a must for anybody whoworks in stochastic processes. It is too bad it is out of print. ... Read more

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From its origins in Newtonian physics, potential theory has developed into a major field of mathematical research. This book provides a comprehensive treatment of classical potential theory: it covers harmonic and subharmonic functions, maximum principles, polynomial expansions, Green functions, potentials and capacity, the Dirichlet problem and boundary integral representations. The first six chapters deal concretely with the basic theory, and include exercises. The final three chapters are more advanced and treat topological ideas specifically created for potential theory, such as the fine topology, the Martin boundary and minimal thinness. The presentation is largely self-contained and is accessible to graduate students, the only prerequisites being a reasonable grounding in analysis and several variables calculus, and a first course in measure theory. The book will prove an essential reference to all those with an interest in potential theory and its applications. ... Read more

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The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincaré inequalities, and spectral synthesis theorems. ... Read more

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This volume contains the revised and completed notes of lectures given at the school "Quantum Potential Theory: Structure and Applications to Physics," held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald from February 26 to March 10, 2007.

Quantum potential theory studies noncommutative (or quantum) analogs of classical potential theory. These lectures provide an introduction to this theory, concentrating on probabilistic potential theory and it quantum analogs, i.e. quantum Markov processes and semigroups, quantum random walks, Dirichlet forms on C* and von Neumann algebras, and boundary theory. Applications to quantum physics, in particular the filtering problem in quantum optics, are also presented.

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Over the past several decades, Italian revolutionary politics has offered a model for new forms of political thinking. Radical Thought in Italy continues that tradition by providing an original view of the potential for a radical democratic politics today that speaks not only to the Italian situation but also to a broadly international context. First, the essays settle accounts with the culture of cynicism, opportunism, and fear that has come to permeate the Left. They then proceed to analyze the new difficulties and possibilities opened by current economic conditions and the crisis of the welfare state. Finally, the authors propose a series of new concepts that are helpful in rethinking revolution for our times. Contributors: Giorgio Agamben, U of Verona and Collège Internationale de Philosophie, Paris; Massimo De Carolis, U of Salerno; Alisa Del Re, U of Padua; Augusto Illuminati, U of Urbino; Maurizio Lazzarato; Antonio Negri, U of Paris VIII; Franco Piperno, U of Calabria; Marco Revelli, U of Turin; Rossana Rossanda; Carlo Vercellone; Adelino Zanini. Paolo Virno is the author of several books, including the recently translated A Grammar of the Multitude. Michael Hardt is professor of literature and romance studies at Duke University. ... Read more

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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.

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This book covers the interplay between the behavior of a class of stochastic processes (random walks) and structure theory. This collection of invited papers by leading researchers presents links with spectral theory and discrete potential theory, besides probabilistic and structure theoretic aspects. Its interdisciplinary approach spans several areas of mathematics including geometric group theory, discrete geometry and harmonic analysis. The work will be of interest to researchers and postgraduate students, both in mathematics and statistical physics. ... Read more

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The aim of the book is to give a unified approach to new developments in discrete potential theory and infinite network theory. The author confines himself to the finite energy case, but this does not result in loss of complexity. On the contrary, the functional analytic machinery may be used in analogy with potential theory on Riemann manifolds.
The book is intended for researchers with interdisciplinary interests in one of the following fields: Markov chains, combinatorial graph theory, network theory, Dirichlet spaces, potential theory, abstract harmonic analysis, theory of boundaries. ... Read more

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A main theme of the book outlines the role of the quantum potential in quantum mechanics and general relativity and one of its origins via fluctuations formulated in terms of Fisher information. Another theme is the description of various approaches to Bohmian mechanics and their role in quantum mechanics and general relativity. Along the way various approaches to, for instance, the Dirac equation, the Einstein equations, the Klein-Gordon equation, the Maxwell equations and the Schrödinger equations are described. Statistics and geometry are intertwined in various ways and, among other matters, the aether, cosmology, entropy, fractals, quantum Kaehler geometry, the vacuum and the zero point field are discussed. There is also some speculative material and some original work along with material extracted from over 1000 references and the work is current up to April 2005. ... Read more

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Bohm's extraordinary potential and modern statistical physics
by B. Roy Frieden

This book is an excellent summary - often in depth -- of modern themes in statistical mathematical physics.Its overall theme is the intimate connection between concepts of quantum fluctuations, expressed via Fisher information, diffusion processes, etc., on one hand, to the so-called "quantum potential" of Bohm on the other.A tremendous range of modern physical theory is presented, of which but a tiny subset can be mentioned in this review.The concept of the quantum potential was championed by D. Bohm as a link between classical and quantum theory, and is a trajectory-based version of elementary quantum mechanics (the trajectories existing at least as a handy "illusion," according to the author.).The Bohm potential has lately been used for similar purposes by Reginatto and Hall, and also to derive an "exact" version of the Heisenberg uncertainty principle.Concepts of information (Shannon- and Fisher- varieties) and entropy are widely used throughout.The Bohm potential is also used in the duality theory of Faraggi and Matoneand in many other concepts of modern quantum theory.The view of physics afforded by the Extreme physical information (EPI) approach of Frieden is objectively and factually summarized.Quantum fluctuation theory is widely used, in application to diffusion theory and stochastic electrodynamics.So also is quantum field theory, in particular work by Nikolic.

This is predominantly a book of mathematical theory, which therefore contains many equations.These are both set out and located within the text.However, the notation is straightforward for any graduate-level theoretical physicist or mathematician, and the writing style is informal and (as a result) informative.There is a wealth of expository material on the aims and accomplishments of the theories. I heartily recommend the book to any physicist or mathematician who wants an in-depth view of today's statistical physics, particularly its tie-ins to the Bohm potential.
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A reprint of the original publication in Grundlehren der mathematischen Wissenschafen, Volume 262. Topics covered include green functions, the Dirichlet problem for relative harmonic functions, the fundamental convergence theorem and the reduction operation, and infirma of families of superharmonic functions. Softcover. ... Read more

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FOUNDATIONS OF POTENTIAL THEORY by OLIVER DIMON KELLOGG. Originally published in 1929.Preface: The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to - the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, in order that the ok may present sound ideals to the student, and also serve the ma pmatician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem Gauss, or Greens theorem on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Pirichlet problem. Exercises are introduced in the conviction that no mastery of a mathematical subject is possible without working with it. They are designed primarily to illustrate or extend the theory, although the desirability of requiring an occasional concrete numerical result has not been lost sight of. O. D. Kellogg. August, 1929. Contents include: Chapter 1. The Force of Gravity. 1. The Subject Matter of Potential Theory 1 2. Newtons Law 2 3. Interpretation of Newtons Law for Continuously Distributed Bodies . 3 4. Forces Due to Special Bodies 4 5. Material Curves, or Wires 8 6 Material Surfaces or Lammas 10 7. Curved Lammas 12 8. Ordinary Bodies, or Volume Distributions 15 9 The Force at Points of the Attracting Masses 17 10. Legitimacy of the Amplified Statement of Newtons Law Attraction between Bodies 22 11. Presence of the Couple Centrobaric Bodies Specific Force 26 Chapter II. Fields of Force. 1. Fields of Force and Other Vector Fields 28 2. Lines of Force 28 3. Velocity Fields 31 4. Expansion, or Divergence of a Field 34 5. The Divergence Theorem 37 6. Flux of Force Solenoidal Fields 40 7. Gauss Integral 42 8. Sources and Sinks 44 9. General Flows of Fluids Equation of Continuity 45 Chapter III The Potential. 1. Work and Potential Energy 48 2 Equipotential Surfaces 54 3. Potentials of Special Distributions 55 4. The Potential of a Homogeneous Circumference 58 5. Two Dimensional Problems The Logarithmic Potential 62 6. Magnetic Particles 65 7. Magnetic Shells, or Double Distributions 66 8. Irrotational Flow 69 . Stokes Theorem 72 10. Flow of Heat 76 11. The Energy of Distributions 79 12... ... Read more

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This book provides an extensive treatment of Potential Theory for sub-Laplacians on stratified Lie groups. It also provides a largely self-contained presentation of stratified Lie groups, and of their Lie algebra of left-invariant vector fields. The presentation is accessible to graduate students and requires no specialized knowledge in algebra or differential geometry.