## Vector and Tensor Analysis with Applications – Borisenko, Tarapov

In this post, we will see the book Vector And Tensor Analysis With Applications by A. I. Borisenko; I. E. Tarapov.

The present book is a freely revised and restyled version of the third edition of the Russian original (Moscow, 1966). As in other volumes of this series, I have not hesitated to introduce a number of pedagogical and mathematical improvements that occurred to me in the course of doing the translation. 1 have also added a brief Bibliography, confined to books in English dealing with approximately the same topics, at about the same level.

The book was translated from Russian by Richard Silverman and was published in 1968.

You can get the book here.

Follow us on The Internet Archive: https://archive.org/details/@mirtitles

Write to us: mirtitles@gmail.com

Fork us at GitLab: https://gitlab.com/mirtitles/

Add new entries to the detailed book catalog here.

# Contents

## Chapter 1 VECTOR ALGEBRA, Page 1.

1.1 Vectors and Scalars, 1.
1.1.1. Free, sliding and bound vectors, 2.

1.2 Operations on Vectors, 3.

1.2.2. Subtraction of vectors, 5.
1.2.3. Projection of a vector onto an axis, 6.
1.2.4. Multiplication of a vector by a scalar, 7.

1.3 Bases and Transformations, 7.
1.3.1. Linear dependence and linear independence of vectors, 7.
1.3.2. Expansion of a vector with respect to other vectors, 8.
1.3.3. Bases and basis vectors, 9.
1.3.4. Direct and inverse transformations of basis vectors, 13.

1.4 Products of Two Vectors, 14.

1.4.1. The scalar product, 14.
1.4.2. The vector product, 16.
1.4.3. Physical examples, 19.

1.5 Products of Three Vectors, 20.

1.5.1. The scalar triple product, 20.
1.5.2. The vector triple product, 21.
1.5.3. “Division” of vectors, 23.

1.6 Reciprocal Bases and Related Topics, 23.
1.6.1. Reciprocal bases, 23.
1.6.2. The summation convention, 26.
1.6.3. Covariant and contravariant components of a vector, 27.
1.6.4. Physical components of a vector, 29.
1.6.5. Relation between covariant and contravariant components, 31.
1.6.6. The case of orthogonal bases, 33.

1.7 Variable Vectors, 35.

1.7.1. Vector functions of a scalar argument, 35.
1.7.2. The derivative of a vector function, 36.
1.7.3. The integral of a vector function, 37.

Solved Problems, 38.
Exercises, 54.

## Chapter 2 THE TENSOR CONCEPT, Page 59.

2.1 Preliminary Remarks, 59.
2.2 Zeroth-Order Tensors (Scalars), 60.

2.3 First-Order Tensors (Vectors), 6l.
2.3.1. Examples, 62.

2.4 Second-Order Tensors, 63.

2.4.1. Examples, 64.
2.4.2. The stress tensor, 66.
2.4.3. The moment of inertia tensor, 68.
2.4.4. The deformation tensor, 70.
2.4.5. The rate of deformation tensor, 72.

2.5 Higher-Order Tensors, 76.
2.6 Transformation of Tensors under Rotations about a Coordinate Axis, 77.
2.7 Invariance of Tensor Equations, 81.

2.8 Curvilinear Coordinates, 82.

2.8.1. Coordinate surfaces, 84.
2.8.2. Coordinate curves, 84.
2.8.3. Bases and coordinates axes, 85.
2.8.4. Arc length. Metric coefficients, 86.

2.9 Tensors in Generalized Coordinate Systems, 88.

2.9.1. Covariant, contravariant and mixed components of a tensor, 88.
2.9.2. The tensor character of giz, g™* and g;*, 89.
2.9.3. Higher-order tensors in generalized coordinates, 90.
2.9.4. Physical components of a tensor. The case of orthogonal bases, 90.
2.9.5. Covariant, contravariant and mixed tensors as such, 91.

Solved Problems, 94.
Exercises, 100.

## 3 TENSOR ALGEBRA, Page 103.

3.2 Multiplication of Tensors, 104.
3.3 Contraction of Tensors, 104.

3.4 Symmetry Properties of Tensors, 105.

3.4.1. Symmetric and antisymmetric tensors, 105.
3.4.2. Equivalence of an antisymmetric second-order tensor to an axial vector, 107.

3.5 Reduction of Tensors to Principal Axes, 109,

3.5.1. Statement of the problem, 109.
3.5.2. The two-dimensional case, 110.
3.5.3. The three-dimensional case, 113.
3.5.4. The tensor ellipsoid, 118.

3.6 Invariants of a Tensor, 121.

3.6.1. A test for tensor character, 122.

3.7 Pseudotensors, 122.

3.7.1. Proper and improper transformations, 122.
3.7.2. Definition of a pseudotensor, 124.
3.7.3. The pseudotensors 125.

Solved Problems, 126.
Exercises, 131.

## Chapter 4 VECTOR AND TENSOR ANALYSIS: RUDIMENTS, Page 134.

4.1 The Field Concept, 134.

4.1.1. Tensor functions of a scalar argurnent, 134.
4.1.2. Tensor fields, 135.
4.1.3. Line integrals. Circulation, 135.

4.2 The Theorems of Gauss, Green and Stokes, 137.

4.2.1. Gauss’ theorem, 137.
4.2.2. Green’s theorem, 139.
4.2.3. Stokes’ theorem, I41.
4.2.4. Simply and multiply connected regions, 144.

4.3 Scalar Fields, 145.

4.3.1. Level surfaces, 145.
4.3.2. The gradient and the directional derivative 146.
4.3.3. Properties of the gradient. The operator 𝛁149.
4.3.4. Another definition of grad 9, 150.

4.4 Vector Fields, 151.

4.4.1. Trajectories of a vector field, 151.
4.4.2. Flux of a vector field, 152.
4.4.3. Divergence of a vector field, 155,
4.4.4. Physical examples, 157.
4.4.5. Curl of a vector field, 161.
4.4.6. Directional derivative of a vector field, 164.

4.5 Second-Order Tensor Fields, 166.

4.6 The Operator V and RelatedDifferential Operators, 168.
4.6.1. Differential operators in orthogonal curvilinear coordinates, 171.
Solved Problems, 174.
Exercises, 182.

## Chapter 5 VECTOR AND TENSOR ANALYSIS: RAMIFICATIONS, Page 185.

5.1 Covariant Differentiation, 185.

5.1.1. Covariant differentiation of vectors, 185.
5.1.2. Christoffel symbols, 187.
5.1.3. Covariant differentiation of tensors, 190.
5.1.4. Ricci’s theorem, 191.
5.1.5. Differential operators in generalized coordinates, 192.

5.2 Integral Theorems, 196.

5.2.1. Theorems related to Gauss’ theorem, 197.
5.2.2. Theorems related to Stokes’ theorem, 198.
5.2.3. Green’s formulas, 201.

5.3 Applications to Fluid Dynamics, 203.

5.3.1. Equations of fluid motion, 203.
5.3.2. The momentum theorem, 208.

5.4 Potential and Irrotational Fields, 211.

5.4.1. Multiple-valued potentials, 213.

5.5 Solenoidal Fields, 216.

5.6 Laplacian Fields, 219.

5.6.1. Harmonic functions, 219.
5.6.2. The Dirichlet and Neumann problems, 222.

5.7 The Fundamental Theorem of Vector Analysis, 223.

5.8 Applications to Electromagnetic Theory, 226.
5.8.1. Maxwell’s equations, 226.
5.8.2. The scalar and vector potentials, 228.
5.8.3. Energy of the electromagnetic field. Poynting’s vector, 230.

Solved Problems, 232.

Exercises 247.

BIBLIOGRAPHY 251.

INDEX 253.