Multilinear Functions Of Direction And Their Uses In Differential Geometry (1921) (Paperback)

MULTILINEAR FUNCTIONS OF DIRECTION AND THEIR USES IN DIFFERENTIAL GEOMETRY BY ERIC HAROLD NEVILLE LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, READING CAMBRIDGE AT THE UNIVERSITY PRESS 1921 PREFACE THE distinctive feature of this work is that the functions dis cussed are primarily not functions of a single variable direction but functions of several independent directions. Functions of a single direction emerge when the directions originally independent become related, and a large number of elementary theorems of differential geometry express in different terms a few properties of a few simple functions since one of the objects of the essay is to emphasise the coordinating power of the theory, the presence of many results with which every reader will be thoroughly familiar calls for no apology. In the applications to the geometry of a single surface two functions thought to be new are described. The first, studied in Section 4, depends on two tangential directions, reduces to normal curvature when these directions coincide, and is called here bilinear curvature. I became acquainted with this function in 1911 and used it in lectures early in 1914. The second, the subject of Section 6, depends on three directions, and reduces to the cubic function as sociated with the name of Laguerre the function is symmetrical, and because the equations of Codazzi can be read as asserting its symmetry I have called the general function the Codazzi function. The theory of multilinear functions does not merely coordinate. It affords simple proofs of the relations between the cubic functions of Laguerre and Darboux 6 231, 6 234 and of formulae 7 242, 7 351, I 7352 for the twist of a family of surfaces, and it leads naturally to expressions 7 241 for the rates of change of the two principal curvatures of a variable member of a family of surfaces at the current point of an orthogonal trajectory of the family, expres sions that are interesting because their existence was deduced by Forsyth in 1903 from an enumeration of invariants. E. H. N. June, 1920. NOTE For the sake of brevity, the space considered is real, but the restriction operates only to the same extent as in other branches of differential geometry. If it is removed, the intrinsic distinction between the positive square root and the negative square root of a given uniform function has to be replaced by a more artificial distinction based on a dissection that is to some extent arbitrary. And there is always a possibility that results need modification if isotropic lines or planes are involved as a rule, nul vectors are admissible as arguments but nul directions are not. CONTENTS PAGE PREFACE ........ . 5 NOTE .......... 6 TABLE OF CONTENTS ....... 7 PRELIMINARY PARAGRAPHS 9 Ol Vectors radials. 0 2 The projected product of two vectors. 0 3 Vector frames. 04 Cartesian axes. 0 5 Directions in a plane. 0-6 Angular differentiation. 1. LINEAR AND MULTILINEAR FUNCTIONS .... 13 Il Definitions of linear and multilinear functions. 1 2 Notation for multilinear functions. 1 3 The core of a multilinear function. 14 The use of reference frames. 1-5 The source of a linear function the projected product of two cores. 1 6 The derivative of a variable core the rate of change of a multi linear function. T7 The gradient of a multilinear function dependent on position in space the rate of change of such afunction along a curve. 1 B The angular derivatives of a multilinear function. 2. FUNDAMENTAL NOTIONS IN THE KINEMATICAL GEO METRY OF SURFACES AND FAMILIES OF SURFACES . 22 2 0 The vectors to be examined. 2 1 Normal curvature and geodesic torsion. 2 2 Geodesic curvature. 2 3 Bilinear curvature the relations of bilinear curvature to normal curvature and geodesic torsion. 24 Swerve geodesic curvature as a swerve. 2-5 The curvature and torsion of a curve on a surface. 2 6 The vectors connected with a family of surfaces...