Differential Geometry by Erwin Kreyszig
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Differential Geometry written by
This book provides an introduction to the differential geometry of curves and surfaces in three-dimensional Euchdean space. Basic concepts and facts of analytic geometry which will be useful for later investigations discused in this textbook. The theory of space curves is presented in the second chapter. We then proceed to the foundations of the theory of surfaces. Problems closely related to the first and second fundamental forms are considered in the third and fourth chapter.
In the theory of surfaces we make full use of the tensor calculus, which is developed as needed, cf. Sections 27-33. The student will quickly find that this calculus becomes a simple tool as soon as he is accustomed to the few basic concepts and rules, especially to the "summation convention’, cf. Section 27. He will perceive that the tensor method is helpful in achieving a simplification of the analytic formalism of many investigations. Hence tensors are important tools in modem differential geometry.
The presentation in this book may also be considered as a preparation for the Riemannian geometry of n dimensions.
As is well known, tensors are of increasing importance not only in mathematics, but also in the application of mathematics to physics and engineering. Since the problems treated in differential geometry by means of tensor calculus are relatively perspicuous, they enable us to understand not only the formalism but also the nature and essential background of this calculus. The student will thus gain by being able to apply his knowledge of tensors to fields other than that of differential geometry. In using tensor calculus one should never forget that the purpose of this calculus lies in its applications to certain problems; it is a tool only, albeit a very powerful one.
Book Detail :-
Title: Differential Geometry
Author(s): Erwin Kreyszig
Publisher: Dover Publications
Series: Dover Books on Mathematics
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About Author :-
Erwin O. Kreyszig (1922–2008) was a German Canadian applied mathematician and the Professor of Mathematics at Carleton University in Ottawa, Ontario, Canada. He was a pioneer in the field of applied mathematics: non-wave replicating linear systems. He was also a distinguished author, having written the textbook Advanced Engineering Mathematics, the leading textbook for civil, mechanical, electrical, and chemical engineering undergraduate engineering mathematics.
Kreyszig received his Ph.D. degree in 1949 at the University of Darmstadt under the supervision of Alwin Walther.
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Book Contents :-
Differential Geometry written by
cover the following topics.
2. THEORY OF CURVES
3. CONCEPT OF A SURFACE. FIRST FUNDAMENTAL FORM. FOUNDATIONS OF TENSOR-CALCULUS
4. SECOND FUNDAMENTAL FORM. GAUSSIAN AND MEAN CURVATURE OF A SURFACE
5. GEODESIC CURVATURE AND GEODESICS
7. ABSOLUTE DIFFERENTIATION AND PARALLEL DISPLACEMENT
8. SPECIAL SURFACES
ANSWERS TO PROBLEMS
COLLECTION OF FORMULAE
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