The Very Basics of Tensors

iUniverse, 2005 - Mathematics - 137 pages
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