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**About this book :- **
**Tensors and Differential Geometry Applied ** written by
** Z. U. A. Warsi**.

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: **
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**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

Gradient, Divergence, Curl, and Laplacian

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

?1

?2

- Abstract Algebra
- Calculus
- Differential Equations
- Engineering Mathematics
- Linear Algebra
- Math Magic
- Real Analysis