Differential geometry of curves and surfaces in E[3] (tensor approach)
著者
書誌事項
Differential geometry of curves and surfaces in E[3] (tensor approach)
Anshan, c2007
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Differential geometry of curves and surfaces in E3 (tensor approach)
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注記
On t.p. "[3]" is superscript
Includes bibliographical references (p. [187]) and index
内容説明・目次
内容説明
This book is intended to suit mathematics courses for post-graduate students. It has been written by an author with 25 years experience of lectures on differential geometry, and is therefore designed to help the reader overcome the difficulties in understanding the underlying concepts of the subject. The book will also be useful for introducing the methodology of differential geometry to research students in associated disciplines; physics, engineering, biosciences and economics. The book is divided into 5 chapters - curvilinear co-ordinates, geometry of space curves, intrinsic geometry of a surface, fundamental formulae of a surface, curves on a surface - and each chapter contains numerous examples which are either worked out or given as an exercise in order to facilitate understanding. Finally the book concludes with a brief history of differential geometry. This book is an excellent text for post-graduate maths courses, and will also be of interest to all mathematicians.
目次
1. CURVILINEAR COORDINATES 1.1Curvilinear Coordinate System in E3 1.2 Elementary Arc Length 1.3 Length of a Vector 1.4 Angle betweenTwo Non-null Vectors 1.5 Reciprocal Base System 1.6 On the Meaning of Covariant Derivatives 1.7 Intrinsic Differentiation 1.8 Parallel Vector Fields 2. GEOMETRY OF SPACE CURVES 2.1 Serret-Frenet Formulae 2.2 Equation of a Straight Line in Curvilinear Coordinate system 2.3 Some Results on Curvature and Torsion. How to Find out Curvature and Torsion of Space Curves 2.4 Helix 3. INTRINSIC GEOMETRY OF A SURFACE 3.1 Curvilinear Coordinates of a Surface 3.2 The Element of Length and the Metric Tensor 3.3 The First Fundamental Form 3.4 Directions on a Surface. Angle between Two Directions 3.5 Geodesic and its Equations 3.6 Parallelism with respect to a Surface 3.7 Intrinsic and Covariant Differentiation of Surface Tensors 3.8 The Riemann-Christoffel Tensor. The Gaussian Curvature of a Surface 3.9 The Geodesic Curvature of a Curve on a Surface 4. THE FUNDAMENTAL FORMULAE OF A SURFACE 4.1 The Tangent Vector to a Surface 4.2 The Normal Vector to a Surface 4.3 The Tensor Derivation of Tensors 4.4 Gauss's Formulae: The second Fundamental Form of a Surface 4.5 Weingarten's Formulae: The Third Fundamental Form of a Surface 4.5 The Equations of Gauss and Codazzi 5. CURVES ON A SURFACE 5.1 The Equations of a Curve on a Surface 5.2 Meusnier's Theorem 5.3 The principal curvatures 5.4 The Lines of Curvature 5.5 The Asymptotic Lines. Enneper's Formula 5.6 The Geodesic Torsion of a Curve on a Surface
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