About the Book
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
5.6.4.1. Initial Concepts
5.6.4.2. Christoffel Symbol
5.6.4.3. Riemann-Christoffel Tensor
5.6.4.4. Ricci Tensor
5.6.4.5. Scalar Curvature
5.6.4.6. 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
About the Author:
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."
