Book description
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...
