**1. Cartesian Tensor Algebra **

1.1 Scalars, Vectors and Tensors

1.2 Index Notations and Cartesian Summation Convention

1.3 Kronecker Delta and Permutation Symbols

1.4 Cartesian Coordinates and Rotation of Axes

1.5 Laws of Transformation of Base Vectors

1.6 Algebra of Vectors

1.7 Second Order Tensors

1.8 Algebra of Cartesian Tensors

1.9 Principal Axes of Second Order Tensors

Exercise 1

**2. Cartesian Tensor calculus**

2.1 Partial Derivatives of Scalar and Vector Field, Gradient, Divergence and Curl

2.2 Gauss’s, Stokes’ and Green’s Theorems in Index Notation

Exercise 2

**3. Rectilinear and curvilinear coordinate systems**

3.1 Rectilinear Coordinate Systems

3.2 The Reciprocal Basis

3.3 Derivation of Formula for Determining Reciprocal Basis

3.4 Curvilinear Coordinate Systems

3.5 Proper Transformations

3.6 Basis and Reciprocal Basis in Curvilinear Coordinate Systems

Exercise 3

**4. General Tensors and the metric Tensor**

4.1 General Tensors

4.2 The Metric Tensor

4.3 The Permutation Tensors

4.4 Tensor Algebra

4.5 The Quotient Rule

4.6 Physical Components of a vector in curvilinear Coordinate systems

4.7 Scalar Product, Vector Product and Scalar Triple Product in Various Forms

Exercises 4

**5. Christoffel symbols and covariant differentiation**

5.1 Partial Derivative of a Vector

5.2 Christoffel Symbols

5.3 Christoffel Symbols in terms of Derivative of Metric Tensors

5.4 Christoffel Symbols in Orthogonal Coordinate Systems

5.5 Transformation of Christoffel Symbols

5.6 Covariant Derivative of Contravariant Components of a Vector

5.7 Covariant Derivative of Covariant Components of a Vector

5.8 Covariant Derivative of Contravariant Components of a Second-Order Tensor

5.9 Covariant Derivative of Covariant Components of a Second-Oder Tensor

5.10 Covariant Derivative of Mixed Components of a Second-Order Tensor, Covariant Derivative of a Scalar

5.11 Laws of Covariant Differentiation, Second Covariant Derivative of Contravariant Components of a Vector

5.12 Ricci’s Theorem

5.13 Gradient of a Scalar Field

5.14 Divergence of a Vector Field

5.15 Curl of a Vector Field

5.16 Laplacian of a Scalar Field

5.17 Laplacian of Vector Field

5.18 Intrinsic Derivative

Exercise 5

6. Application of Tensors in fluid dynamics 183

6.1 Equation of continuity

6.2 Expression for Acceleration of a Fluid Particle in Euclidean Space

6.3 Expression for the Covariant Derivative of Physical Components of the Velocity Vector

6.4 Cauchy’s Eqation of Motion in Orthogonal Curvilinear Coordinates

6.5 Some Useful Results of Vector Calculus

Exercise 6

Index 199

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