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E-raamat: Introduction to Tensor Analysis

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  • Formaat: 126 pages
  • Ilmumisaeg: 01-Sep-2022
  • Kirjastus: River Publishers
  • Keel: eng
  • ISBN-13: 9781000795912
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Introduction to Tensor AnalysisSuurem pilt
  • Formaat: 126 pages
  • Ilmumisaeg: 01-Sep-2022
  • Kirjastus: River Publishers
  • Keel: eng
  • ISBN-13: 9781000795912
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The subject of Tensor Analysis deals with the problem of the formulation of the relation between various entities in forms which remain invariant when we pass from one system of coordinates to another. The invariant form of equation is necessarily related to the possible system of coordinates with reference to which the equation remains invariant. The primary purpose of this book is the study of the invariance form of equation relative to the totality of the rectangular coordinate system in the three-dimensional Euclidean space. We start with the consideration of the way the sets representing various entities are transformed when we pass from onesystem of rectangular coordinates to another. A Tensor may be a physicalentity that can be described as a Tensor only with respect to the manner of itsrepresentation by means of multi-sux sets associated with different systems of axes such that the sets associated with different systems of coordinates obey thetransformation law for Tensor. We have employed sux notation for tensors of anyorder, we could also employ single letter such as A,B to denote Tensors.

This is a short introduction to the topic of Tensor Analysis. A tensor is an entity which is represented in any coordinate system by an array of numbers called its components. The components change from coordinate system to coordinate in a systematic way described by rules. The arrays of numbers are not the tensor; they are only the representation of the tensor in a particular coordinate system. The special properties of tensors are important for solving problems in Physics and Geometry.
Preface v
Syllabus vii
1 Introduction
1(6)
1.1 Symbols Multi-Suffix
2(1)
1.2 Summation Convention
3(4)
2 Cartesian Tensor
7(32)
2.1 Introduction
7(1)
2.2 Transformation of Coordinates
8(2)
2.3 Relations Between the Direction Cosines
10(1)
2.4 Transformation of Velocity Components
11(1)
2.5 First-Order Tensors
12(1)
2.6 Second-Order Tensors
13(1)
2.7 Notation for Tensors
14(1)
2.8 Algebraic Operations on Tensors
14(3)
2.8.1 Sum and Difference of Tensors
15(1)
2.8.2 Product of Tensors
16(1)
2.9 Quotient Law of Tensors
17(2)
2.10 Contraction Theorem
19(2)
2.11 Symmetric and Skew-Symmetric Tensor
21(2)
2.12 Alternate Tensor
23(1)
2.13 Kronecker Tensor
24(1)
2.14 Relation Between Alternate and Kronecker Tensors
25(1)
2.15 Matrices and Tensors of First and Second Orders
26(2)
2.16 Product of Two Matrices
28(3)
2.17 Scalar and Vector Inner Product
31(1)
2.17.1 Two Vectors
31(1)
2.17.2 Scalar Product
31(1)
2.17.3 Vector Product
31(1)
2.18 Tensor Fields
32(3)
2.18.1 Gradient of Tensor Field
32(2)
2.18.2 Divergence of Vector Point Function
34(1)
2.18.3 Curl of Vector Point Function
34(1)
2.19 Tensorial Formulation of Gauss's Theorem
35(1)
2.20 Tensorial Formulation of Stoke's Theorem
35(1)
2.21 Exercise
36(3)
3 Tensor in Physics
39(16)
3.1 Kinematics of Single Particle
39(2)
3.1.1 Momentum
40(1)
3.1.2 Acceleration
40(1)
3.1.3 Force
40(1)
3.2 Kinetic Energy and Potential Energy
41(1)
3.3 Work Function and Potential Energy
41(2)
3.4 Momentum and Angular Momentum
43(1)
3.5 Moment of Inertia
44(2)
3.6 Strain Tensor at Any Point
46(3)
3.7 Stress Tensor at any Point P
49(1)
3.7.1 Normal Stress
50(1)
3.7.2 Simple Stress
50(1)
3.7.3 Shearing Stress
50(1)
3.8 Generalised Hooke's Law
50(1)
3.9 Isotropic Tensor
51(1)
3.10 Exercises
52(3)
4 Tensor in Analytic Solid Geometry
55(12)
4.1 Vector as Directed Line Segments
55(2)
4.2 Geometrical Interpretation of the Sum of two Vectors
57(1)
4.3 Length and Angle between Two Vectors
57(1)
4.4 Geometrical Interpretation of Scalar and Vector Products
58(3)
4.4.1 Scalar Triple Product
60(1)
4.4.2 Vector Triple Products
60(1)
4.5 Tensor Formulation of Analytical Solid Geometry
61(3)
4.5.1 Distance Between Two Points P(xi) and Q(yi)
61(1)
4.5.2 Angle Between Two Lines with Direction Cosines
61(1)
4.5.3 The Equation of Plane
62(1)
4.5.4 Condition for Two Line Coplanar
63(1)
4.6 Exercises
64(3)
5 General Tensor
67(18)
5.1 Curvilinear Coordinates
68(1)
5.2 Coordinate Transformation Equation
68(1)
5.3 Contravariant and Covariant Tensor
69(2)
5.4 Contravariant Vector or Contravariant Tensor of Order-One
71(1)
5.5 Covariant Vector or Covariant Tensor of Order-One
71(1)
5.6 Mixed Second-Order Tensor
72(1)
5.7 General Tensor of Any Order
72(1)
5.8 Metric Tensor
73(1)
5.9 Associate Contravariant Metric Tensor
74(1)
5.10 Associate Metric Tensor
75(1)
5.11 Christoffel Symbols of the First and Second - Kind
76(3)
5.12 Covariant Derivative of a Covariant Vector
79(1)
5.13 Covariant Derivative of a Contravariant Vector
80(1)
5.14 Exercises
81(4)
6 Tensor in Relativity
85(14)
6.1 Special Theory of Relativity
85(3)
6.2 Four-Vectors in Relativity
88(3)
6.3 Maxwell's Equations
91(3)
6.4 General Theory of Relativity
94(1)
6.5 Spherically Symmetrical Metric
95(1)
6.6 Planetary Motion
96(1)
6.7 Exercises
97(2)
7 Geodesies and Its Coordinate
99(10)
7.1 Families of Curves
99(1)
7.2 Euler's Form
100(1)
7.3 Geodesies
101(2)
7.4 Geodesic Form of the Line Elements
103(2)
7.5 Geodesic Coordinate
105(2)
7.6 Exercise
107(2)
Index 109(2)
About the Authors 111
Bipin Singh Koranga, Sanjay Kumar Padaliya
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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