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Appendix a Tensor Mathematics

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Appendix a Tensor Mathematics Appendix A Tensor Mathematics A.l Introduction In physics we are accustomed to relating quantities to other quantities by means of mathematical expressions. The quantities that represent physical properties of a point or of an infinitesimal volume in space may be of a single numerical value without direction, such as temperature, mass, speed, dist­ ance, specific gravity, energy, etc.; they are defined by a single magnitude and we call them scalars. Other physical quantities have direction as well and their magnitude is represented by an array of numerical values, their number corresponding to the number of dimensions of the space, three numerical values in a three-dimensional space; such quantities are called vectors, for example velocity, acceleration, force, etc., and the three numerical values are the components of the vector in the direction of the coordinates of the space. Still other physical quantities such as moment of inertia, stress, strain, permeability coefficients, electric flux, electromagnetic flux, etc., are repre­ sented in a three-dimensional space by nine numerical quantities, called components, and are known as tensors. We will introduce a slightly different definition. We shall call the scalars tensors of order zero, vectors - tensors of order one and tensors with nine components - tensors of order two. There are, of course, in physics and mathematics, tensors of higher order. Tensors of order three have an array of 27 components and tensors of order four have 81 components and so on. Couple stresses that arise in materials with polar forces are examples of tensors of order three, and the Riemann curvature tensor that appears in geodesics is an example of a tensor of order four. The coefficients of an elastic crystal also form a tensor of the fourth order with 81 components, but owing to isotropy and to symmetry in geometry and in energies they are soon reduced to the two known coefficients. In general we can say that a tensor of order n in a three-dimensional space will have 3n components. Why tensor mathematics? For two reasons. First, a shorthand presentation 492 Appendix A Tensor Mathematics of the equations is possible, since one equation identically replaces three or more equations in any other mathematical notation, thus affording a great simplification in the mathematics. Second, any mathematical equation pre­ sented is valid for any coordinate system, the rules of transformation from one coordinate system to another being predetermined. This appendix contains not only the basics of tensor analysis, which is necessary to follow the exposition of this book, but also additional material to enable the reader to continue his studies, perhaps in other directions. For further insight and more detailed derivations a great selection of publications is available, among them, Synge and Schild (1949), Lass (1950), Schouten (1965), Spain (1953), Ericksen (1960a), Borisenko and Tarapov (1968). A.2 The Indicial Notation If we are given a set of N independent variables, say, coordinates x, y, z, ... , N, we find it more convenient to denote them by the same letter, distinguishing between them by means of indices. Thus we shall write the N variables Xl> X2, X3, ••• , XN or Xj, Xj' Xb ••• , XN, or it may be written more compactly x" where r takes in turn the values of 1, 2, 3, ... , or of i, j, k, ... , up to N. In the same way that we write the index r as a subscript, we can use superscripts instead, by writing x'. The italic characters used for superscripts and subscripts distinguish them from power exponents, which are roman characters. Non-tensorial indices, denoting generic groups of expressions or numericals of summation, will be marked by gothic indices, while the use of greek indices is reserved for tensorial summations, as will be explained later. If we have a three variable system or a three manifold coordinate system or triad, (N = 1, 2, 3 or N = X, y, z), then x', Xj, xj, Xq represent notations for tensors of the first order, that is, vectors, in a tridimensional triad, where x and X are the respective values of the components of the vectors in the directions r, i, j, q, which may take in turn the values of 1, 2, 3 or x, y, z, or of any other triad. A.3 Transformation of Coordinates Let us consider a point in an N dimensional space, defined by the coordin­ ates of its position vector, Xl, X 2 , X 3 , ••• , XN. The N equations (A.3.1) The Summation Convention 493 where JC are single-valued continuous differentiable functions of the coordin­ ates, define a transformation of coordinates into a new coordinate system Xi. The necessary and sufficient condition that the N equations (A.3.1) be independent is satisfied if the determinant formed from the partial derivatives axijaxj , named Jacobian, does not vanish ax! ax! ax! -- -- -- ax! ax2 axN ax 2 ax 2 ax 2 a(!X ,x 2 ,x 3 , ... , x N) J= -- -- -- *0 ax! ax2 axN a(x!, x 2, x 3, ... , XN) axN axN axN -- -- -- ax! ax2 axN (A.3.2) Conversely, Eq. (A.3.1) may be solved for the JC as a function of Xi . '! 2 N X' = X'(x ,x , ... , x ) (A.3.3) In a tridimensional coordinate system, where N = 3, Eq. (A.3.2) becomes ax! ax! ax! -- -- -- ax! ax2 ax3 ax 2 ax 2 ax 2 a(x!, x2, x3) J= -- -- -- - *0 (A.3.4) axl ax2 ax3 a(xl, x 2, X 3) ax 3 ax 3 ax 3 -- axl ax2 ax3 As our study is concerned with tridimensional systems, we shall restrict ourselves, from here on, to tridimensional coordinates, unless otherwise stated. A.4 The Summation Convention An italic index appearing twice in a term implies a summation. As a convention, we shall transpose such repeated indices to Greek indices in 494 Appendix A Tensor Mathematics order to stress the fact that the indices are no longer tensorial indices but "dummy" indices. For instance, the expression N S = AIXI + A2X2 + ... + ANxN = L AiXi = A",x'" (A.4.1) n=l and the total differential of Xi from Eq. (A.3.1) (A.4.2) contain summations over the Greek indices. The expression A "'''' represents a summation of the identically indexed components of a second-order tensor A,s (A.4.3) We shall introduce two conventions: Range convention. A free unrepeated italic index will have the range of values 1,2,3. Summation convention. A repeated Greek index is a dummy index, i.e. a result of transposed repeated italic indices, and is to be summed from 1 to 3. A summation reduces by two the tensorial order of the term in which it appears. A.S The Kronecker Delta The Kronecker delta c5~, also known as the unit tensor, is defined so that its components equal zero whenever i -=1= j, and 1 if i = j. Its matricial form is 1 o o lc5jl = 0 1 o (A.5.1) o o 1 The obvious property of the unit tensor is that when it is multiplied by a tensor of any order it maintains the tensor intact, c5~A '" = Ai. Also axk/axj = c5j and c5~ = 3. Contravariant and Covariant Tensors 495 A.6 Contravariant and Covariant Tensors The components of a vector Ai, a first -order tensor, are said to be components of a contravariant tensor if by changing coordinates from Xi to Xi they transform according to equation (A.6.1) Conversely, by multiplying Eq. (A.6.1) by axk/axi and summing over i, we obtain (A.6.2) From Eq. (A.6.2) follows the transformation (A.6.3) The components of a vector Ai' are said to be components of a covariant tensor if they transform at the change of the coordinate X to the coordinate Xi, according to equation (A.6.4) Similarly to Eq. (A.6.2), by multiplying Eq. (A.6.4) by axi/axk and summing over the index i from 1 to 3, we obtain ax'" ax'" ax{3 ax'" --a =----A =--A =A (A.6.5) axk '" axk ax'" {3 axk '" k The term of/aX which forms a first-order tensor from a scalar function f which is a zero-order tensor, will, in any other coordinate system, have the components of ax'" of (A.6.6) Such a covariant tensor is called the gradient of f. A second-order contravariant tensor A ij transforms according to Eq. (A.6.1) (A.6.7) 496 Appendix A Tensor Mathematics Similarly, a covariant second-order tensor Aij will transform, according to Eq. (A.6.4), as follows ax a ax f3 aij = --,---,- Aaf3 (A.6.8) ax' ax' A second-order tensor whose components are A}, will transform (A.6.9) A} are the components of a mixed tensor. Higher-order tensors transform according to the above rules as well. The components of a fifth-order mixed tensor A%lm, for instance, will transform as follows (A.6.10) The order of transformation is important. Finally, it should be noted that the coordinates JC, Xi do not form components of a contravariant tensor, although they seem to suggest it by appearance. A most important deduction from Eq. (A.6.10) is that if all components of a tensor are zero in one coordinate system, they are zero in every other coordinate system as well. A.7 Symmetric and Skew-symmetric Tensors The order of indices in a tensor is meaningful. The tensor A ij is not necessarily the same as the tensor Aji. Tensor A ij may be called the transpose of tensor Aji, a name borrowed from the mathematics of matrices.
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