1.4 LECTURE 4. Tensors and Vector Identities

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1 16 CHAPTER 1. VECTOR ALGEBRA Triple Product The triple product of three vectors A, B and C is defined by In tensor notation it is A ( B C ) = [ A, B, C ] = A ( B C ) i, j,k=1 ε i jk A i B j C k = A 1 A 2 A 3 B 1 B 2 B 3 C 1 C 2 C 3 It is equal to the signed volume of a parallelepiped based on three vectors A i, B j, C k Properties. 1. [A, B, C] = [B, C, A] = [C, A, B] = [B, A, C] = [C, B, A] = [A, C, B], 2. [A, B, C] = 0 if and only if the vectors are coplanar, 3. [A, B, C] is linear in each argument, 4. for an orthonormal basis [ e 1, e 2, e 3 ] = LECTURE 4. Tensors and Vector Identities We will denote the Cartesian coordinates by x 1 = x, x 2 = y, x 3 = z and the unit vectors in the direction of positive axes (called the standard basis vectors) by e 1 = i, e 2 = j, e 3 = k This can be denoted simply by x i and e j, where i, j = 1, 2, 3. For the indices one usually uses the lowercase Latin letters i, j, k, l, m, n etc. (do not confuse with i, j, k). If you run out of letters, you can use any other letters. The convention is though that the indices are denoted by small (versus capital) Latin (versus Greek) letters, and take values 1, 2, 3. Greek indices are used in four-dimensional space-time in special relativity, where they take values 0, 1, 2, 3, with x 0 = t denoting time. vecanal332.tex; August 25, 2017; 16:29; p. 16

2 1.4. LECTURE 4. TENSORS AND VECTOR IDENTITIES 17 The scalar products of the basis vectors are: { 1, if i = j e i e j = 0, if i j One says that they form an orthonormal system. This can be written in a compact form by defining so called Kronecker symbol δ i j { 1, if i = j δ i j = 0, if i j This can also be represented by the unit 3 3 matrix (δ i j ) = Then e i e j = δ i j Physical quantities, like mass, energy, volume, temperature, density etc., that can be described by one number are called scalars. This number does not depend on the coordinate system; it is an invariant. Vectors are physical quantities, like velocity, position, displacement, force, acceleration, electric field, magnetic field etc., that are described by three numbers. A tensor is a geometric object that requires for its full description more than just one number, as scalar, and even more than three numbers, as a vector. Examples of tensors include: stress tensor, strain tensor, inertia tensor, energymomentum tensor, tensor of the electromagnetic field, metric tensor, curvature tensor etc. These numbers are called the components of the tensor. The components of a tensor are labeled by indices, for example, δ i j, ε i jk, T i j, B i j, σ i j, R i i jk A tensor whose all components are zero is called a zero tensor. The tenswors with upper indices are called contravariant, and the ones with lower indices are called covariant. If a tensor has both types of indices vecanal332.tex; August 25, 2017; 16:29; p. 17

3 18 CHAPTER 1. VECTOR ALGEBRA then it is of mixed type. The total number of indices is called the rank of the tensor. A tensor that has p upper indices and q lower indices T i 1...i p j1... j q is called a tensor of type (p, q); it has the rank r = p + q. So, a scalar is a tensor of rank 0, a vector is a tensor of rank 1 etc. The actual numerical values of the components of a tensor do depend on the coordinate system. If one changes the coordinate system, for example, rotates it, then the components of a tensor will change. If one goes from the Cartesian coordinate system to a curvilinear coordinate system, for example, a system of spherical or cylindrical coordinates, then the components of a tensor will also change. It is this transformation law of the components of the tensor that makes a collection of numbers a tensor. We will not give the formal definition of a tensor, rather we give here a very short review of tensor analysis in Cartesian coordinates along with some very useful formulas and rules that enable one to deal with tensors. In any tensor equation an index can appear only once (single index) or twice (repeated index). A pair of repeated indices cannot appear more than once. Einstein Summation Convention. One always encounters the sums over the indices that appear twice in an equation. According to the standard convention, called Einstein summation convention, one has agreed to sum over repeated indices and omit the summation signs. For example, δ i j A i B j = i=1 δ i j A i B j j=1 A i B i = δ i i = ε i jk A j C k B i = i=1 A i B i i=1 i=1 δ i i j=1 k=1 ε i jk A j C k B i vecanal332.tex; August 25, 2017; 16:29; p. 18

4 1.4. LECTURE 4. TENSORS AND VECTOR IDENTITIES 19 One can add tensors of the same type. The result is a tensor of the same type. One can multiply tensors by scalars. The result is a tensor of the same type. If one multiplies a tensor of rank r with a tensor of rank k, one gets a new tensor of rank r + k. More precisely, if one multiplies a tensor of type (p, q) with a tensor of type (r, s), then one gets a new tensor of type (p + r, q + s). For example, A i B j = C i j, T mn σ i j = R mn i j Note: one just multiplies the components of the tensors without any summation. Given a tensor of type (p, q) (that is of rank r = p + q) one may select a pair of indices, of which one should be an upper index and another an lower index, and replace them by two identical (repeated indices), summation over the latter being implied by the summation convention. This process is called contraction. As a result one gets a new tensor of type (p 1, q 1) of rank r 2 = p + q 2. For example, A i i, R i j ki, C ik k (1.4.3) Clearly, δ i i = δ1 1 + δ2 2 + δ3 3 = 3 (1.4.4) A tensor of rank 2 is said to be symmetric if A i j = A ji (1.4.5) and anti-symmetric (or skew-symmetric) if A i j = A ji (1.4.6) Any tensor A i j of second rank can be decomposed A i j = A (i j) + A [i j] (1.4.7) into its symmetric and anti-symmetric parts A (i j) = 1 2 (A i j + A ji ) (1.4.8) A [i j] = 1 2 (A i j A ji ) (1.4.9) vecanal332.tex; August 25, 2017; 16:29; p. 19

5 20 CHAPTER 1. VECTOR ALGEBRA One can also symmetrize a tensor of rank 3 over three indices: B (i jk) = 1 6 (B i jk + B jki + B ki j + B ik j + B jik + B k ji ) (1.4.10) Correspondingly, the anti-symmetrization of a tensor of rank 3 is defined by B [i jk] = 1 6 (B i jk + B jki + B ki j B ik j B jik B k ji ) (1.4.11) What one does here is one sums over all possible permutations of indices and changes sign if the permutation is odd. Remark. The contraction of symmetric and an anti-symmetric tensors is equal to zero. Let a tensor A i j be symmetric and B i j be antisymmetric. Then A i j B i j = A ji B i j = A ji B ji = A i j B i j, (1.4.12) and, therefore, A i j B i j = 0. (1.4.13) The scalar products of the basis vectors e i define a symmetric second rank tensor called the metric tensor e i e j = g i j. The contravariant components of the metric tensor are defined by the inverse matrix g i j = (g i j ) 1 In Cartesian coordinates the components of the metric tensor are given by Kronecker delta symbol g i j = δ i j, g i j = δ i j. The metric tensor can be used to raise and lower indices of tensors. For example, if A i are contravariant components of a vector then its covariant components are A i = g i j A j Conversely, A i = g i j A j This operations, called raising and lowering indices can be applied to any tensor. vecanal332.tex; August 25, 2017; 16:29; p. 20

6 1.4. LECTURE 4. TENSORS AND VECTOR IDENTITIES 21 The scalar product of two vectors is A B = g i j A i B j. Remark. In Cartesian coordinates the covariant and contravariant components are equal A i = A i. In Cartesian coordinates the position of the tensor indices (up or down) does not make any difference. Therefore, δ i j A i B j = A i B i = A B Levi-Civita symbol ε i jk is defined by +1, if (i, j, k) is an even permutation of (1, 2, 3) ε i jk = 1, if (i, j, k) is an odd permutation of (1, 2, 3) 0, otherwise If one raises the indices then one sees that in Cartesian coordinates one obtains the same symbol, so that ε i jk = ε i jk (1.4.14) The Levi-Civita symbol defines a tensor of rank 3 (strictly speaking it is a pseudo-tensor density of weight 1), called a Levi-Civita tensor. It describes the signed volume of a parallelepiped based on three displacement vectors A i, B j, C k V = ε i jka i B j C k The Levi-Civita symbol defines a completely antisymmetric tensor. It changes sign under the permutation of any two indices. ε i jk = ε jik = ε k ji = ε ik j ε i jk = ε jki = ε ki j As a consequence its contraction vanishes ε i j j = 0, also, for any vector A i ε i jk A j A k = 0 vecanal332.tex; August 25, 2017; 16:29; p. 21

7 22 CHAPTER 1. VECTOR ALGEBRA Further, one can show that the product of two Levi-Civita symbols can be expressed in terms of the Kronecker symbols ε i jk ε mnl = 6δ m [i δn j δl k] = δ m i δn j δl k + δm j δn k δl i + δm k δn i δl j δ m i δn k δl j δm j δn i δl k δm k δn j δl i By contracting the indices k and l we get ε i jk ε mnk = 2δ m [i δn j] = δ m i δn j δm j δn i further, by contracting the indices j and n we obtain ε i jk ε m jk = 2δ m i and, finally, by contracting the indices i and m we have ε i jk ε i jk = 6 The vector product D = B C in tensor notation is given by D i = ( B C ) i = ε i jk B j C k The triple product is then [ A, B, C ] = A ( B C ) = ε i jk A i B j C k Note that the position of indices (up versus down) in Cartesian coordinates is not important. However, it is still more clear, when you see one index up and the same index down then you should immediately notice that this is a contraction and there is a summation over this index from 1 to 3. We repeat once again that the name of such repeated indices is not important, they are dummy indices; one can rename them to any other letter if needed (make sure that there are no other indices with that name in the given tensor equation!). vecanal332.tex; August 25, 2017; 16:29; p. 22

8 1.4. LECTURE 4. TENSORS AND VECTOR IDENTITIES Vector Identities Tensor notation is very useful in vector analysis, in particular when manipulating the multiple vector products and vector identities. By using the properties of Levi-Civita symbol and Kronecker symbol one can derive now all vector identities. For example, A ( B C ) = B ( A C ) C ( A B ) ( A B ) ( C D ) = [ D, A, B ] C [ C, A, B ] D ( A B ) ( C D ) = (A C)(B D) (A D)(B C) A ( B C ) + B ( C A ) + C ( A B ) = 0 Proofs. [( A B ) ( C D )] i = ε i jk ( A B ) j ( C D ) k = ε i jk ε jmn A m B n ε kpq C p D q = (δ p i δq j δp j δq i )ε jmn A m B n C p D q = (δ p i εqmn δ q i εpmn )A m B n C p D q = ε qmn A m B n C i D q ε pmn A m B n C p D i = [ D, A, B ]C i [ C, A, B ]D i vecanal332.tex; August 25, 2017; 16:29; p. 23