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A Treatise On The Differential Geometry Of Curves And Surfaces (Hardcover)
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A Treatise On The Differential Geometry Of Curves And Surfaces (Hardcover)
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A TREATISE ON THE DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES.
PREFACE: This book is a development from courses which I have given
in Princeton for a number of years. During this time I have come to
feel that more would be accomplished by my students if they had an
introductory treatise written in English and otherwise adapted to
the use of men beginning their graduate work. Chapter I is devoted
to the theory of twisted curves, the method in general being that
which is usually followed in discussions of this subject. But in
addition I have introduced the idea of moving axes, and have
derived the formulas pertaining thereto from the previously
obtained Freiiet-Serret fornmlas. In this way the student is made
familiar with a method which is similar to that used by Darboux in
the tirst volume of his Lepons, and to that of Cesaro in his
Gcomctria Ittiriiiseca. This method is not only of great advantage
in the treat ment of certain topics and in the solution of
problems, but it is valu able iu developing geometrical thinking.
The remainder of the book may be divided into threo parts. The
iirst, consisting of Chapters II-VI, deals with the geometry of a
sur face in the neighborhood of a point and the developments
therefrom, such as curves and systems of curves defined by
differential equa tions. To a large extent the method is that of
Gauss, by which the properties of a surface are derived from the
discussion of two qxiad ratie differential forms. However, little
or no space is given to the algebraic treatment of differential
forms and their invariants. In addition, the method of moving axes,
as defined in the first chapter, has been extended so as to be
applicable to an investigation of theproperties of surf ac. es and
groups of surfaces. The extent of the theory concerning ordinary
points is so great that no attempt has been made to consider the
exceptional problems. Por a discussion of uch questions as the
existence of integrals of differential equa tions and boundary
conditions the reader must consult the treatises which deal
particularly with these subjects. lu Chapters VII and VIII the
theory previously developed is applied to several groups of
surfaces, such as the quadrics, ruled surfaces, minimal surfaces,
surfaces of constant total curvature, and surfaces with plane and
spherical lines of curvature The idea of applicability of surfaces
is introduced in Chapter IIT as a particular case of conformal
representation, and throughout the book attention is called to
examples of applicable surfaces. However, the general problems
concerned with the applicability of surfaces are discussed in
Chapters IX and X, the latter of which deals entirely with the
recent method of Weingarten and its developments. The remaining
four chapters are devoted to a discussion of infinitesimal
deformation of surfaces, congruences of straight Hues and of
circles, and triply orthogonal systems of surfaces. It will be
noticed that the book contains many examples, and the student will
find that whereas certain of them are merely direct applications of
the formulas, others constitute extensions of the theory which
might properly be included as portions of a more ex tensive
treatise. At first I felt constrained to give such references as
would enable the reader to consult the journals and treatises from
which some of these problems were taken, but finally it seemed best
to furnish, no such key, only to remark that the flncyklopadie der
mathematisc7ien Wissensckaften may be of assistance. And the same
may be said about references to the sources of the subject-matter
of the book. Many important citations have been made, but there has
not been an attempt to give every reference. However, I desire to
acknowledge niy indebtedness to the treatises of Uarboux, Biancln,
and Scheffers...
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