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### Tensors and Differential Geometry Applied by Z. U. A. Warsi

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The purpose of this monograph is to present the theories of basic tensor analysis and of the differential geometry of surfaces for the purpose of formulating problems of coordinate generation in regions bounded by arbitrary curves or surfaces. Since the writing of the first memoir on the subject of tensor analysis by Ricci and Levi-Civita [11 in 1901 some very significant developments in the theory of tensor analysis have taken place, though, the major applications of the subject have only been confined to the general theory of relativity and to the continuum mechanics. In this monograph an attempt has been made to utilize the theories of classical tensor analysis and differential geometry of surfaces in developing new methods for the generation of coordinates in arbitrary regions. Only those results of tensor theoretic and differential geometric significance have been explained which are needed in the development of the subject in a fruitful manner. However, it turns out that for a better understanding and a sound conceptual orientation some basic ideas, by the way of definitions and notations, have also to be introduced. Though this elementary exposition forms a small part of the total effort, and is explained much better in the references given below, nevertheless, its inclusion imparts a sort of continuity to the whole presentation.

Book Detail :-
Title: Tensors and Differential Geometry Applied
Edition:
Author(s): Z. U. A. Warsi
Publisher:
Series:
Year: 1981
Pages: 212
Type: PDF
Language: Englsih
ISBN:
Country: Join WhatsApp Group Join Telegram Group

Book Contents :-
Tensors and Differential Geometry Applied written by Z. U. A. Warsi cover the following topics. '
**Part-I Fundamental Concepts and Basic Tensor Forms
Preliminaries
Summation Convention on Cartesian Components, Vector Multiplication Using Cartesiaj Components, Placement of Indices (Covariant and Contravariant), Dyads, Operation with Dyads, Curvilinear Coordinates, Various Representations in Terms of a i and ai, Differential Operations in Curvilinear Coordinates, Gradient of Vectors and Divergence of Tensors
Euclidean and Riemannian Spaces
Fundamental Tensor Structures and Transformation Laws, Relations Between the Base Vectors, Transformation Laws for Vectors and Tensors, Algebraic Properties of Tensors
Differentiation of Vectors and Tensors
Christoffel Symbols: Their Properties and Transformation Laws
Transformation Laws for Christoffel Symbols, Formulae: Cartesian to Curvilinear and Vice Versa
Miscellaneous Derivations
The Curvature Tensor and Its Implications
Algebra of the Curvature Tensor, The Possibility of Local Cartesian Coordinates in a Riemannian Space, Ricci's Tensor, Bianchi's Identity, A Divergence-Free Tensor
The Geometry of the Event-Space
Newtonian Mechanics Using the Principles of Special Relativity, Application to the Navier-Stokes Equations
**Part-II The Geometry of Curves and Surfaces
Theory of Curves
Serret-Frenet Equations
Geometry of Two-Dimensional Surfaces Embedded in E3
Normal Curvature of a Surface: Second Fundamental Form, Principal Normal Curvatures, Equations for the Derivatives of Surface Normal (Weingarten Equations), Formulae of Gauss and the Surface Christoffel Symbols, Christoffel Symbols, Intrinsic Nature of the Gaussian Curvature (Equations of Codazzi and Mainardi), A Particular Form of Codazzi Equations, The Third Fundamental Form, The Geodesic Curvature, Geodesics and Parallelism on a Surface, Differential Parameters of Beltrami, First Differential Parameters
Mapping of Surfaces
Isothermic and Equiareal Coordinates on a Sphere
Some Standard Parametric Representations
**Part-III Basic Differential Models for Coordinate Generation
Problem Formulation
Collection of Some Useful Expansions and Notation
Differential Equations for Coordinate Generation Based on the Riemann Tensor
Laplacians of 4, r and C and Their Inversions, Laplacians in Orthogonal Coordinates, Riemann Curvature Tensor for Specitic Surfaces, Coordinates in a Plane, Determination of the Cartesian Coordinates, Coordinate Generation Capabilities of the Developed Equations, Two-Dimensional Orthogonal Coordinates in a Plane, Three-Dimensional Orthogonal Coordinates
Differential Equations for Coordinate Generation Based on the Formulae of Gauss
Formulation of the Problem, Particular Case of Eqs. (81)-(83). (Minimal Surfaces), Coordinate Generation Between Two Prescribed Surfaces, Coordinate Redistribution, An Analytical Example of Coordinate Generation
Appendix 1. Christoffel Symbols in Three-Dimensional Coordinates
Appendix 2. Christoffel Symbols Based on Surface Coordinates
Appendix 3. The Beltrami Equations
Bibliography
Index

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