This textbook provides a rigorous approach to tensor manifolds in several aspects relevant for Engineers and Physicists working in industry or academia. With a thorough, comprehensive, and unified presentation, this book offers insights into several topics of tensor analysis, which covers all aspects of n-dimensional spaces.
The main purpose of this book is to give a self-contained yet simple, correct and comprehensive mathematical explanation of tensor calculus for undergraduate and graduate students and for professionals. In addition to many worked problems, this book features a selection of examples, solved step by step.
Although no emphasis is placed on special and particular problems of Engineering or Physics, the text covers the fundamentals of these fields of science. The book makes a brief introduction into the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having the grounds for the subsequent themes, without needing to resort to other bibliographical sources on tensors.
Chapter 1 deals with Fundamental Concepts about tensors and chapter 2 is devoted to the study of covariant, absolute and contravariant derivatives. The chapters 3 and 4 are dedicated to the Integral Theorems and Differential Operators, respectively. Chapter 5 deals with Riemann Spaces, and finally the chapter 6 presents a concise study of the Parallelism of Vectors. It also shows how to solve various problems of several particular manifolds.
Prof. Dr. Emil de Souza Sanchez Filho received his degree in civil engineering in 1976, his M. Sc. in 1998, and his D. Sc. in 1992 from COPPE-Federal University of Rio de Janeiro. In 1993-1994 he became visiting Researcher at Technische Universitat Braunschweig , Germany. In the periods 2003-2005 and 2013-2014 he worked as a postdoctoral fellow at PUC-Rio, where he is today Guest Professor (2004-2015). Currently, the author is full professor at Fluminense Federal University, Brazil. He worked in the construction industry and has large experience in Structural Engineering. He has published more than 220 papers (articles in peer reviewed journals and conference articles), chapters in 9 books, and acted as editor of 3 books and as author of 3 books: Solid Mechanics Elements, Tensors, and Tensor Calculus.
Chapter 1. Fundamental Concepts 1.1. Basic Concepts 1.2. N Dimensional Spaces 1.3. Homogeneous and Isotropic Spaces 1.4. Kronecker Delta 1.5. Metric Tensor 1.6. Angle between Curves 1.7. Some Useful Formulas Chapter 2. Covariant, Absolute and Contravariant Differentiation 2.1. Initial Notes 2.2. Cartesian Tensor Differentiation 2.3. Base Vectors Differentiation< 2.4. Christoffel Symbols 2.5. Covariant Differentiation 2.5.1. Contravariant Tensor 2.5.2. Covariant Tensor 2.5.3. Mixed Tensor 2.5.4. Covariant Differentiation: Addition and Product of Tensors 2.5.5. Covariant Differentiation of the Tensors 2.5.6. Particularities of the Covariant Derivative 2.6. Covariant Differentiation of the Relative Tensors 2.7. Intrinsic or Absolute Differentiation 2.8. Contravariant Differentiation Chapter 3. Integral Theorems 3.1. Initial Concepts 3.2. Green Theorem 3.3. Stokes Theorem 3.4. Gauss-Ostrogadsky Theorem Chapter 4. Differential Operators 4.1. Scalar, Vectorial and Tensorial Fields 4.2. Gradient 4.3. Divergent 4.4 Curl 4.5. Successive Applications of the Nabla Operator 4.5.1 Basic Relations 4.5.2 Laplace Operator 4.5.3 Other Differential Operators Chapter 5. Riemann Spaces 5.1. Initial Notes 5.2. Curvature of the Space 5.3. Riemann Curvature 5.4. Ricci Tensor and Scalar Curvature 5.5. Einstein Tensor 5.6. Particular Cases of Riemann Spaces 5.6.1. Riemann Space 5.6.2. Riemann Space with Constant Curvature 5.6.3. Minkowski Space 5.6.4. Conformal Spaces 22.214.171.124. Initial Concepts 126.96.36.199. Christoffel Symbol 188.8.131.52. Riemann-Christoffel Tensor 184.108.40.206. Ricci Tensor 220.127.116.11. Scalar Curvature 18.104.22.168. Weyl Tensor 5.7. Dimensional Analysis Chapter 6. Parallelisms of Vectors 6.1. Initial Notes 6.2. Geodesics 6.3. Null Geodesics 6.4. Coordinates Systems 6.4.1. Geodesic Coordinates 6.4.2. Riemann Coordinates 6.5. Geodesic Deviation 6.6. Parallelism of Vectors 6.6.1. Initial Notes 6.6.2. Parallel Transport of Vectors 6.6.3. Torsion