Название: Differential Geometry and Its Visualization
Автор: Eberhard Malkowsky, Cemal Dolicanin, and Vesna Velickovic
Издательство: CRC Press
Год: 2024
Страниц: 493
Язык: английский
Формат: pdf (true)
Размер: 23.3 MB

Differential Geometry and Its Visualization is suitable for graduate level courses in differential geometry, serving both students and teachers. It can also be used as a supplementary reference for research in mathematics and the natural and engineering sciences.

Differential geometry is the study of geometric objects and their properties using the methods of mathematical analysis. The classical theory of curves and surfaces in three-dimensional Euclidean space is presented in the first three chapters. The abstract and modern topics of tensor algebra, Riemannian spaces and tensor analysis are studied in the last two chapters. A great number of illustrating examples, visualizations and genuine figures created by the authors’ own software are included to support the understanding of the presented concepts and results, and to develop an adequate perception of the shapes of geometric objects, their properties and the relations between them.

This book is intended for students and teachers of courses in differential geometry at graduate level as well as in doctoral studies. It is also useful for mathematicians, physicists and engineers who need or are interested in its topics. The material of the book is presented in a comprehensive and self-contained way and strongly supported by a great number of examples and visualizations.

The prerequisites are a solid background from the undergraduate courses in analytical geometry and analysis. As a reminder, the most important results from these areas are listed in Section 1.2.

The book contains five chapters with a total of 54 sections including 215 figures. Major parts of the first three chapters are translations from Serbian of the authors’ book Diferencijalna geometrija i njena vizuelizacija.

Chapter 1 contains the study of the basic local geometric properties of curves in three-dimensional Euclidean E3. The topics of special interest include the parametric representation of curves, the arc length of a curve as its natural parameter, many examples of planar curves and the geometrical principles of their constructions, the frame or the vectors of the trihedra of curves, Frenet’s formulae, the curvature and torsion of curves and their geometric significance, characteristic shape of a curve in a neighbourhood of any of its points, osculating circles and spheres, involutes and evolutes of curves, the fundamental theorem of curves and the natural or intrinsic equations of curves, lines of constant slope and finally spherical images of curves.

Chapter 2 deals with the local differential geometry of surfaces in three-dimensional Euclidean space E3. This means the study of the geometric shape of surfaces in the neighbourhood of an arbitrary one of their points. The most important concepts arising in this task are those of the normal, principal, Gaussian and mean curvature.

Chapter 3 deals with the intrinsic geometry of surfaces, that is, with the geometric properties of surfaces that only depend on measurements on the surface itself, namely on its first fundamental coefficients and their derivatives.

Chapter 4 deals with the redevelopment of the intrinsic geometry of surfaces independently of their embedding in three-dimensional space, based only on the definition of measurements of lengths in a point set.

Features:

Extensive, full colour visualisations
Numerous exercises
Self-contained and comprehensive treatment of the topic