The Classical Field Theories

PDF methods for turbulent reactive flows

This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Diffusions, Markov processes, and martingales

The application of integral equation methods to exterior boundary-value problems for Laplace’s equation and for the Helmholtz (or reduced wave) equation is discussed. In the latter case the straightforward formulation in terms of a single integral equation may give rise to difficulties of non-uniqueness; it is shown that uniqueness can be restored by deriving a second integral equation and suitably combining it with the first. Finally, an outline is given of methods for transforming the integral operators with strongly singular kernels which occur in the second equation.

The application of integral equation methods to the numerical solution of some exterior boundary-value problems

Linear elasticity is one of the more successful theories of mathematical physics Its pragmatic success in describing the small deformations of many materials is uncontested The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson The theoretical development of the subject continued at a brisk pace until the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff, Lame, Saint-Venant, Somigliana, Stokes, and others These authors established the basic theorems of the theory, namely compatibility, reciprocity, and uniqueness, and deduced important general solutions of the underlying field equations In the 20th century the emphasis shifted to the solution of boundary-value problems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Venant’s principle, stress functions, variational principles, and uniqueness

The Linear Theory of Elasticity

Foundations of potential theory

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Harmonic functions and Green’s integral

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Invariant points in function space

Let U be harmonic in a closed region R, whose boundary conltains a regular surface element E, with a representation z=f(x, y). If E has bounded curvatures, and if O(x, y) and the boundary values of U on E have continuous derivatives of order n which satisfy a Dini condition, then the partial derivatives of U of order n exist, as limits, on E, and are continuous in R at any interior point of E. H6lder conditions on the boundary values of U, or on their derivatives of order n, imply Holder conditions on U, or the corresponding derivatives, in R, in the neighborhood of the interior points of E.

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On the derivatives of harmonic functions on the boundary

The next step in gaining an insight into the character of Newtonian attraction will be to think of the forces at all points of space as a whole, rather than to fix attention on the forces at isolated points. When a force is defined at every point of space, or at every point of a portion of space, we have what is known as a field of force. Thus, an attracting body determines a field of force. Analytically, a force field amounts to three functions (the components of the force) of three variables (the coordinates of the point).

Oliver Dimon Kellogg

Papers

Foundations of potential theory

Harmonic functions and Green’s integral

Invariant points in function space

On the derivatives of harmonic functions on the boundary

Fields of Force