Morris Kline

A note on the expansion coefficient of geometrical optics

Abstract : The expansion coefficient of geometrical optics is a measure of the cross section os, New York U., N. Y. A NOTE ON THE EXPANSION COEFFICIENT OF GEOMETRICAL OPTICS, by Morris Kline. 1961, 12p. (Research rept. no. EM-166) (Contract AF 19(604)5238) Unclassified report DESCRIPTORS: *Optics, *Electromagnetic theory, *Differential geometry, Light, Electron optics, Series, Geometry. The expansion coefficient of geometrical optics is a measure of the cross section of a tube of rays and has the physical significance of measuring the intensity of the light propagating along the tube. Strictly, it is a point concept and measures the intensity along an individual ray. This paper presents a convenient expression for the expansion coefficient. The mathematics involved is merely an application of known differential geometry but the expression derived seems to be new and is apparently unknown to workers in electromagnetic theory. The new feature of this paper is that the formula given for the variation of the expansion coefficient holds in inhomogeneous isotropic media and reduces immediately to the widely known expression for the expansion coefficient in homogeneous media, which involves the Gauss curvature of the wave front. (Author)

Freshman Mathematics as an Integral Part of Western Culture

1. Introduction. I would like to reopen the very hackneyed question of what to teach the liberal arts student. My excuse for doing this is that the answer which has been given for generations is equally hackneyed and, what is more to the point, a totally unsatisfactory one. Let me state at the outset that by the liberal arts student I mean the one who does not intend to use mathematics in some profession or career, and is taking mathematics because the subject is supposed to contribute to a liberal education. Though much might be said about the proper freshman courses for students who intend to continue their mathematical training beyond this first year or who may take just one year of mathematics but will have to apply it in specialized physics, chemistry, and biology courses (e.g., pre-medical students), the needs of these latter two groups of students will not be discussed here. The recommendations to be made in this article concerning the liberal arts students are based on the assumption that this group can be segregated from the others. This segregation can be effected even in small colleges without creating any serious administrative problems. As defined above, the liberal arts students constitute only one group of freshmen. However, by far the greatest percentage of freshmen belongs to this group. Also, since the segregation that is presupposed is in accordance with interests rather than ability, the liberal arts group will contain some of the most worthwhile students. For these reasons, then, this group must be given the utmost consideration. Moreover, since many of these students will become leaders in our society they will determine the fate of mathematics in some areas. Hence there are selfish reasons too for being concerned about the knowledge and impressions of mathematics which these students will acquire.

Representation of homeomorphisms in Hilbert space

where ya is the a th coordinate of the transformed point and xp the /3th coordinate of the original point, and a runs through all the coordinate indices. However, the methods of imbedding 5 whereby such a representation of ƒ becomes possible generally do not permit us to simplify the dimension or structure of the cartesian space when 5 is more specialized. Tychonoff has shown* that every normal f topological space 5 with a neighborhood system of power less than or equal to r can be imbedded in (a bicompact part of) a cartesian space RT in which each point has r real numbers as coordinates. In particular, if r = ^0, S can be imbedded in the fundamental parallelopiped of Hilbert space.

Mathematical Thought from Ancient to Modern Times

Now it is possible to refine Tychonoffs method slightly so that a homeomorphism g of S onto itself can be represented by equations of the form (1) acting in RT. And when S has a countable neighborhood system, the equations (1) will act in the fundamental parallelopiped of Hilbert space. Thus, to represent a homeomorphism of these more restricted spaces by (1) we can confine ourselves to the fundamental parallelopiped of Hilbert space.