The Very Basics of Tensors

Front Cover
iUniverse, 2005 - Mathematics - 137 pages
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Tensor calculus is a generalization of vector calculus, and comes near of being a universal language in physics. Physical laws must be independent of any particular coordinate system used in describing them. This requirement leads to tensor calculus. The only prerequisites for reading this book are a familiarity with calculus (including vector calculus) and linear algebra, and some knowledge of differential equations.
 

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Contents

Spaces points and coordinates
1
Distance between adjacent points in a Euclidean 2space for example an ordinary plane a Euclidean 3space for example ordinary space and a Euclidea...
2
Flat and curved spaces
3
Transformation of coordinates in Euclidean 3space Curvilinear coordinates
4
Orthogonal curvilinear coordinates
5
Vectors in Euclidean 3space with general curvilinear coordinates
6
Contravariant and covariant components of a vector in Euclidean 3space
10
The relation between the contravariant components of a vector in two general curvilinear coordinate systems in Euclidean 3space
11
How to perform calculations
46
The line element and the metric tensor Riemannian space
49
The Jacobian of coordinate transformations in Euclidean 3space
53
Area and volume in Euclidean 3space
55
The Kronecker delta the Kronecker tensor
61
The conjugate metric tensor the reciprocal metric tensor
63
Associated tensors Raising and lowering indices
66
The physical components of a vector Ap or A
70

The relation between the covariant components of a vector in two general curvilinear coordinate systems in Euclidean 3space
14
Proof of the result in section 8 Generalization to TVspace
16
Proof of the result in section 9
19
General differentiable TVspaces and general coordinates Curves and surfaces
23
Contravariant tensors of rank one
25
Covariant tensors of rank one
28
Mixed tensors of the second rank
30
Contravariant and covariant tensors of rank two
31
Tensors of rank greater than two
33
Invariants or scalars tensors of rank zero
34
Tensor equations
35
Tensor fields
36
Addition of tensors
37
Subtraction of tensors
38
Outer Multiplication of tensors
39
Division of tensors
40
Contraction of tensors
41
Inner multiplication of tensors
42
Quotient law of tensors
43
Symmetry
44
Skewsymmetry or antisymmetry
45
The physical components of a tensor Apq or A
72
Distance in a Riemannian space
74
Angle between two curves in a Riemannian space
76
Length of a vector in a Riemannian space
78
Angle between two vectors in a Riemannian space
79
Notes on spaces coordinates tensors and physics
80
Partial derivative ordinary derivative of a tensor in general Aspace
82
The Lie derivative of a general tensor field
83
The affine connection and covariant differentiation
89
The intrinsic absolute derivative and affine geodesies
99
The RiemannChristoffel tensor
102
Affine flatness
105
Christoffels symbols
107
Metric geodesies
112
The metric connection the covariant derivative and the absolute derivativ
121
Metric flatness
123
A few easy examples from physics in a general Riemannian Aspace with general coordinates
124
A few easy exercises
129
Index The numbers refer to sections
131
Selected Bibliography
136
Copyright

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About the author (2005)

Oeijord, a graduate of the Agricultural University of Norway, also studied mathematics at the University of Trondheim, in Norway as well. He is a former assistant professor of mathematics at Tromsoe College, Norway, and is the author of several scientific works in Norwegian. He is currently a full time science writer.

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