Oliver Dimon Kellogg

On the derivatives of harmonic functions on the boundary

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.

The Logarithmic Potential

We have seen in Chapter VI, § 7 (p. 172), that logarithmic potentials are limiting forms of Newtonian potentials. We have seen also that harmonic functions in two dimensions, being special cases of harmonic functions in space, in that they are independent of one coordinate, partake of the properties of harmonic functions in space. The only essential differences arise from a change in the definition of regularity at infinity, and the character of these differences has been amply illustrated in the exercises at the close of Chapter IX (p. 248).

Converses of Gauss’ theorem on the arithmetic mean

1.

The Force of Gravity

While the theory of Newtonian potentials has various aspects, it is best introduced as a body of results on the properties of forces which are characterized by Newtons Law of Universal Gravitation1: Every particle of matter in the universe attracts every other particle, with a force whose direction is that of the line joining the two, and whose magnitude is directly as the product of their masses, and inversely as the square of their distance from each other. If, however, potential theory were restricted in its applications to problems in gravitation alone, it could not hold the important place which it does, not only in mathematical physics, but in pure mathematics as well. In the physical world, we meet with forces of the same character acting between electric charges, and between the poles of magnets. But as we proceed, it will become evident that potential theory may also be regarded as the theory of a certain differential equation, known as LAPLACE’S. This differential equation characterizes the steady flow of heat in homogeneous media, it characterizes the steady flow of ideal fluids, of steady electric currents, and it occurs fundamentally in the study of the equilibrium of elastic solids.

Sequences of Harmonic Functions

We have already found need of the fact that certain infinite series of harmonic functions converge to limiting functions which are harmonic. We are now in a position to study questions of this sort more system- atically. Among the most useful is the following theorem due to HAR- NACK

The Divergence Theorem

We have already seen something of the role of the divergence theorem and of Stokes’ theorem in the study of fields of force and other vector fields; we shall also find them indispensable tools in later work. Our first task will be to prove them under rather restrictive assumptions, so that the proofs will not have their essential features buried in the minutiae which are unescapable if general results are to be attained.

Properties of Newtonian Potentials at Points Occupied by Masses

We continue our study of the properties of Newtonian potentials, now in the neighborhood of points of the distributions of matter. Our object is to find relations between the potential and the density, for the purpose indicated at the beginning of the last chapter. As it is only in the neighborhood of a point of a distribution that the density at that point makes itself felt in a preponderating way, we must of necessity investigate the behavior of the potentials at such points.

Fields of Force

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).

Potentials as Solutions of Laplace’s Equation; Electrostatics

The fundamental law of electrostatics was discovered by Couloumb1, and states that the force between two small charged bodies is proportional in magnitude to the product of the charges and inversely proportional to the square of their distance apart,