Tensor Calculus for Engineers and Physicists

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

2.5.5. Covariant Differentiation of the  Tensors 

2.5.6. Particularities of the Covariant Derivative

2.6. Covariant Differentiation of the Relative Tensors

2.8. Contravariant Differentiation

3.1. Initial Concepts

3.2. Green Theorem

3.3. Stokes 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

5.6.4.4.  Ricci Tensor

5.6.4.5. Scalar Curvature

5.6.4.6. Weyl Tensor

5.7. Dimensional Analysis            

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