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Über dieses Buch

Let us first state exactly what this book is and what it is not. It is a compendium of equations for the physicist and the engineer working with electrostatics, magne­ tostatics, electric currents, electromagnetic fields, heat flow, gravitation, diffusion, optics, or acoustics. It tabulates the properties of 40 coordinate systems, states the Laplace and Helmholtz equations in each coordinate system, and gives the separation equations and their solutions. But it is not a textbook and it does not cover relativistic and quantum phenomena. The history of classical physics may be regarded as an interplay between two ideas, the concept of action-at-a-distance and the concept of a field. Newton's equation of universal gravitation, for instance, implies action-at-a-distance. The same form of equation was employed by COULOMB to express the force between charged particles. AMPERE and GAUSS extended this idea to the phenomenological action between currents. In 1867, LUDVIG LORENZ formulated electrodynamics as retarded action-at-a-distance. At almost the same time, MAXWELL presented the alternative formulation in terms of fields. In most cases, the field approach has shown itself to be the more powerful.

Inhaltsverzeichnis

Frontmatter

Section I. Eleven Coordinate Systems

Abstract
The book is limited to orthogonal coordinate systems in euclidean 3-space. Skew coordinate systems do not allow separation of variables and will not be considered.
Parry Moon, Domina Eberle Spencer

Section II. Transformations in the Complex Plane

Abstract
The most promising way of extending the engineering applications of field theory is to develop new coordinate systems. Section I listed the eleven systems whose coordinate surfaces are of the first or second degree. Klein [12] and Bôcher [4] extended this list to include a class of fourth-degree surfaces known as cyclides [13]. All possible systems of this class are treated by Bôcher. They include the eleven coordinates of Section I, as well as more complicated coordinates; but all are either simply separable or R-separable (Section IV).
Parry Moon, Domina Eberle Spencer

Section III. Cylindrical Systems

Abstract
The cylindrical coordinate systems are obtained by translating each of the maps, Figs. 2.01 to 2.21, in a direction perpendicular to the graphs, thus forming two orthogonal families of cylinders. The third family of coordinate surfaces consists of parallel planes, z = const. The coordinate axis that is parallel to the generators of the cylinders is called the z-axis in all cases.
Parry Moon, Domina Eberle Spencer

Section IV. Rotational Systems

Abstract
Each of the 21 transformations of Section II yields one or two rotational coordinate systems, obtained by twirling the plane map about an axis. A tabulation of the most interesting of these systems is given in this section.
Parry Moon, Domina Eberle Spencer

Section V. The Vector Helmholtz Equation

Abstract
The vector Helmholtz equation, which occurs particularly in electromagnetic theory [19], is more complicated than the scalar Helmholtz equation and its separation presents new problems.
Parry Moon, Domina Eberle Spencer

Section VI. Differential Equations

Abstract
Previous sections have treated the separation of the Laplace and Helmholtz equations in 40 coordinate systems. In this section, all the separation equations are tabulated in a systematic manner. Each equation is designated in terms of its singularities in the complex plane, and the general solutions of the differential equations are listed.
Parry Moon, Domina Eberle Spencer

Section VII. Functions

Abstract
The purpose of this section is to provide a summary of the mathematical functions obtained as solutions of the differential equations of field theory. The summary is necessarily incomplete, since many of the functions have never been thoroughly investigated. In particular, almost nothing is known about the properties of the various wave functions. Recent tabulation of spheroidal wave functions is a beginning in this direction, but much remains to be done.
Parry Moon, Domina Eberle Spencer

Backmatter

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