VECTORS, TENSORS AND INDEX NOTATION

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1 VECTORS, TENSORS AND INDEX NOTATION Enrico Nobile Dipartimento di Ingegneria e Architettura Università degli Studi di Trieste, TRIESTE March 5, 2018 Vectors & Tensors, E. Nobile March 5, / 36 OUTLINE 1 Vectors The scalar product The vector product The scalar triple product The vector triple product The Nabla operator 2 Index notation Summation convention and dummy index Free index The Kronecker delta Rules of index notation The Permutation Symbol Gradient, divergence and curl Vectors & Tensors, E. Nobile March 5, / 36

2 Review of vector algebra The scalar product The scalar product of two vectors a and b (dot product) is usually written a b, and is defined as the scalar quantity: a b = a b cos θ Some properties of the scalar product a (b + c) = a b + a c a a = a 2 a b = (a x i + a y j + a z k) (b x i + b y j + b z k) = a x b x + a y b y + a z b z Vectors & Tensors, E. Nobile March 5, / 36 Review of vector algebra - cont. The vector product The vector product of two vectors a and b (cross product) is usually written a b (or a b), and is defined as the vector quantity a b = a b sin θ n Some properties of the vector product b a = a b a (b + c) = a b + a c a b = (a x i + a y j + a z k) (b x i + b y j + b z k) = (a y b z a z b y ) i + (a z b x a x b z ) j + (a x b y a y b x ) k a b = i j k a x a y a z b x b y b z Vectors & Tensors, E. Nobile March 5, / 36

3 Review of vector algebra - cont. Vector area A A = a b = a b sin θ n Volume of a parallelepiped V = c (a b) = a (b c) = b (c a) Vectors & Tensors, E. Nobile March 5, / 36 Scalar triple product We already saw that a (b c) = b (c a) = c (a b) A useful way of expressing the scalar triple product is in determinant form. Since i j k a b = a x a y a z b x b y b z it follows that Scalar triple product a (b c) = a x a y a z b x b y b z c x c y c z Vectors & Tensors, E. Nobile March 5, / 36

4 Vector triple product Geometric interpretation From the properties of the cross-product, it follows that the vector a (b c) is perpendicular both to a and to (b c). Since (b c) is perpendicular to both b and c, it follows that a (b c) must lie in the plane containing both b and c (see figure). Therefore we may write a (b c) = αb + βc where α and β are yet to be determined. This can be done evaluating the vector-product on the left-hand side, and use the right-hand side to determine α and β. Vectors & Tensors, E. Nobile March 5, / 36 Vector triple product - cont. If a = a x i + a y j + a z k, and analogously for b and c, then b c = i (b y c z b z c y ) j (b x c z c x b z ) + k (b x c y c x b y ) Using the determinant form of the vector product i j k a (b c) = a x a y a z b y c z b z c y c x b z b x c z b x c y c x b y = i [a y (b x c y c x b y ) a z (c x b z b x c z )] j [a x (b x c y c x b y ) a z (b y c z b z c y )] + k [a x (c x b z b x c z ) a y (b y c z b z c y )] Regrouping according to the form αb + βc a (b c) = ib x (a y c y + a z c z ) + jb y (a x c x + a z c z ) + kb z (a x c x + a y c y ) jc x (a y b y + a z b z ) jc y (a x b x + a z b z ) kc z (a x b x + a y b y ) Vectors & Tensors, E. Nobile March 5, / 36

5 Vector triple product - cont. Adding and subtracting i (b x a x c x ) + j (b y a y c y ) + k (b z a z c z ) to the previous expression finally gives a (b c) = (a c) b (a b) c (1) and therefore α = a c β = a b Vectors & Tensors, E. Nobile March 5, / 36 Gradient of a scalar A vector operator, which is frequently used in fluid dynamics and heat transfer, is the nabla, (or del) operator, defined as = x i + y j + k When the nabla operator is applied on a scalar variable φ, it results in the gradient of φ, given by: φ = φ x i + φ y j + φ k Therefore, the gradient of a scalar field is a vector field and it indicates that the value of φ changes in space in both magnitude and direction. Vectors & Tensors, E. Nobile March 5, / 36

6 Gradient of a scalar - cont. The projection of φ in the direction of the unit vector e l is given by dφ dl = φ e l = φ cos ( φ, e l ) and is called directional derivative of φ along the direction of the unit vector e l. The maximum value of the directional derivative is φ and is achieved when cos( φ, e l ) = 1, e.g. in the direction of φ. Therefore, the gradient of a scalar field φ indicates the direction and magnitude of the largest variation of φ at every point in space. Finally, φ is normal to the φ = const. surface that passes through that point. Vectors & Tensors, E. Nobile March 5, / 36 Operations on the operator The scalar product of the operator with a vector u of components u, v and w in the x, y and z direction, respectively, is the Divergence of the vector u, which is a scalar quantity written as u = u x + u y + u The divergence of the gradient of a scalar variable φ is denoted by the Laplacian operator and is a scalar given by ( φ) = 2 φ = 2 φ x + 2 φ 2 y φ 2 The Laplacian of a vector follows from the above definition and is a vector given by ( ) ( ) ( ) 2 u = 2 u i + 2 v j + 2 w k Vectors & Tensors, E. Nobile March 5, / 36

7 Operations on the operator - cont. Another quantity of interest is the curl of a vector field u, which is a vector obtained by the cross product of the operator and vector u ( u = x i + y j + ) k (ui + vj + wk) i j k = x y u v z ( w = y v ) ( u i + w ) ( v j + x x u ) k y Vectors & Tensors, E. Nobile March 5, / 36 The dyadic product A dyad is a tensor of order two and rank one, and is the result of the dyadic product (also called outer product or tensor product) of two vectors. If a and b are two vectors, then the dyadic product is indicated as ab, a b or ab T, where T means transpose. The dyadic product of two vectors a and b is a Tensor and is given, in matrix form, by a 1 a 1 b 1 a 1 b 2 a 1 b 3 ab a b ab T = a 2 [b 1 b 2 b 3 ] = a 2 b 1 a 2 b 2 a 2 b 3 a 3 a 3 b 1 a 3 b 2 a 3 b 3 Vectors & Tensors, E. Nobile March 5, / 36

8 The vector product of a vector and a tensor The Vector Product (or Dot Product) of a vector a and a tensor T is a vector and is given by a T ( a T T ) T T 11 T 12 T 13 T = [a 1 a 2 a 3 ] T 21 T 22 T 23 T 31 T 32 T 33 = [a 1 T 11 + a 2 T 21 + a 3 T 31 a 1 T 12 + a 2 T 22 + a 3 T 32 a 1 T 13 + a 2 T 23 + a 3 T 33 ] T a 1 T 11 + a 2 T 21 + a 3 T 31 = a 1 T 12 + a 2 T 22 + a 3 T 32 a 1 T 13 + a 2 T 23 + a 3 T 33 Vectors & Tensors, E. Nobile March 5, / 36 A common mixed product An important mixed product, which is frequently found in fluid mechanics, is the following ( [(u ) u] = (ui + vj + wk) x i + y j + ) k (ui + vj + wk) ( = u x + v y + w ) (ui + vj + wk) = = ( u u x + v u y + w u u u x + v u y + w u u v x + v v y + w v u w x + v w y + w w ) ( i + u v x + v v y + w v ) ( j + u w x + v w y + w w ) k Vectors & Tensors, E. Nobile March 5, / 36

9 A common mixed product - cont. It is easy to see that the same result can be obtained considering the vector (dot) product of the vector u with the tensor obtained with the dyadic product of the divergence with the vector u: u ( u) ( u T ( u) ) T = [u v w] u v w x x x u v w y y y u v w [ = u u x + v u y + w u u v x + v v y + w v u w x + v w y + w w = u u x + v u y + w u u v x + v v y + w v u w x + v w y + w w T ] T Vectors & Tensors, E. Nobile March 5, / 36 Index notation: overview The notation adopted to describe vectors is perhaps the most widely accepted. There are a number of alternative notations, and one of the most useful is the index notations, also called Cartesian notation. The index notation is a powerful tool for manipulating multidimensional equations, and it enjoys a number of advantages in comparison to the traditional vector notation: Conciseness: it allows standard results to be written in an immediately clear though compact form. Quick proof of the results that we need. There are other advantages - and a couple of disadvantages - but in addition it is important to note that the index approach is the most natural notation when dealing with a generalization of the vector: the tensor. We will find convenient to relabel the coordinates (x, y, z) as (x 1, x 2, x 3 ) or simply as x i, with the agreement that the index i can take any of the values 1, 2 and 3. Using this index notation, a vector a may be denoted by a i. Therefore the components of the vector a i are (a 1, a 2, a 3 ). There are times when the more conventional vector notation is more useful. It is therefore important to be able to easily convert back and forth between the two. Vectors & Tensors, E. Nobile March 5, / 36

10 Equality and summation convention Two vectors a i and b i are equal if and only if Equality a i = b i (2) If the index is unspecified then the equation in which it occurs is assumed to be valid for each of the three possible values that the index can take, e.g. (2) is shorthand for a 1 = b 1, a 2 = b 2, a 3 = b 3 (3) The summation convention If an index is repeated in a term, then it is assumed that the term is summed as the repeated index takes the values 1, 2 and 3. Thus a i b i means 3 a i b i a 1 b 1 + a 2 b 2 + a 3 b 3 (4) i=1 Using this convention, if a i and b i are two vectors (representing a and b), then the scalar product a b is represented by a i b i. Vectors & Tensors, E. Nobile March 5, / 36 Dummy index Dummy index A repeated index is called a dummy index, since the precise letter used is irrelevant: a i b i has the same meaning as a j b j, a p b p, a s b s etc., that is a 1 b 1 + a 2 b 2 + a 3 b 3. A dummy index appears twice within an additive term of an expression. In the equation below, j and k are both dummy indices a i = ɛ ijk b j c k + D ij e j The dummy index is local to an individual additive term. It may be renamed in one term - so long as the renaming doesn t conflict with other indices - and it does not need to be renamed in other terms (and, in fact, may not necessarily even be present in other terms). Vectors & Tensors, E. Nobile March 5, / 36

11 The summation convention: examples λ = a i b i λ = c i = S ik x k c i = τ = S ij S ij τ = C ij = A ik B kj C ij = C ij = A ki B kj C ij = 3 a i b i λ = a 1 b 1 + a 2 b 2 + a 3 b 3 i=1 3 S ik x k k=1 3 i=1 c 1 = S 11 x 1 + S 12 x 2 + S 13 x 3 c 2 = S 21 x 1 + S 22 x 2 + S 23 x 3 c 3 = S 31 x 1 + S 32 x 2 + S 33 x 3 3 S ij S ij τ = S 11 S 11 + S 12 S S 32 S 32 + S 33 S 33 j=1 3 A ik B kj [C] = [A] [B] k=1 3 A ki B kj [C] = [A] T [B] k=1 Vectors & Tensors, E. Nobile March 5, / 36 Free index Free index An index that is not repeated (in a term) is called a free index. In the equation below, i is a free index: a i = ɛ ijk b j c k + D ij e j The same letter must be used for the free index in every additive term. The free index may be renamed if and only if it is renamed in every term. Terms in an expression may have more than one free index so long as the indices are distinct. For example the vector-notation expression is written A = B T A ij = (B ij ) T = B ji in index notation. This expression implies nine distinct equations, since i and j are both free indices. Vectors & Tensors, E. Nobile March 5, / 36

12 Free index - cont. The number of free indices in a term equals the rank of the term: Notation Rank scalar a 0 vector a i 1 tensor A ij 2 tensor A ijk 3 Technically, a scalar is a tensor with rank 0, and a vector is a tensor of rank 1. Tensors may assume a rank of any integer greater than or equal to zero. You may only sum together terms with equal rank. The first free index in a term corresponds to the row, and the second corresponds to the column. Thus, a vector (which has only one free index) is written as a column of three rows a = a i = a 1 a 2 a 3 and a rank-2 tensor is written as A = A ij = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 Vectors & Tensors, E. Nobile March 5, / 36 The Kronecker delta We define a two-indexed quantity δ ij called the Kronecker delta by δ ij = { 1 if i = j 0 otherwise (5) Thus δ 11 = δ 22 = δ 33 = 1 and δ 12 = δ 21 = δ 13 = δ 31 = δ 23 = δ 32 = 0. Another name for this quantity is the substitution operator, since it can be used to substitute one index for another. This follows from the easiliy verified result a j = δ ji a i (6) The right-hand side is δ j1 a 1 + δ j2 a 2 + δ j3 a 3, so if j = 1 in (6) a 1 = δ 11 a 1 + δ 12 a 2 + δ 13 a 3 = a 1, with similar results for j = 2 and j = 3. Like in ordinary algebra, an equation will be formed from a collection of terms added together and equated to zero. Each term will include one or more indexed objects multiplied together. Vectors & Tensors, E. Nobile March 5, / 36

13 Rules 1 The same index (subscript) may not appear more than twice in a product of two (or more) vectors or tensors. Thus A ik x k, A ik B kj, A ij B ik C nk are valid, but are meaningless. A kk x k, A ik B kk, A ij B ik C ik 2 The number and type of free indices in any term of a given equation must be the same. Thus are valid, but are meaningless. x i = u i + c i x = u + c a i = A ki B kj x j + C ik u k a = A T Bx + Cu x i = A ij x j = A ik u k x i = A ik u k + c j Vectors & Tensors, E. Nobile March 5, / 36 Rules - cont. 3 Free and dummy indices may be changed without altering the meaning of an expression, provided that rules 1 and 2 are not violated. Thus x i = A ik x k x j = A jk x k x j = A ji x i Vectors & Tensors, E. Nobile March 5, / 36

14 Examples Example 1 If a i and b i correspond to vectors a and b, the index form of the vector equation a + (a b)b = 0 is the following a i + a j b j b i = 0 Example 2 Verify that δ ij δ ij = 3 δ ij δ ij = δ ij δ ij i j δ ij δ ij = 3 = δ 11 δ 11 + δ 12 δ 12 + δ 13 δ 13 + δ 21 δ 21 + δ 22 δ 22 + δ 32 δ 32 + δ 31 δ 31 + δ 32 δ 32 + δ 33 δ 33 Vectors & Tensors, E. Nobile March 5, / 36 Permutation Symbol The Permutation Symbol or Levi-Civita symbol 1, sometimes also called Alternating Unit Tensor, is defined by 1 if i, j, k is a cyclic permutation of 1, 2, 3 ɛ ijk = 1 if i, j, k is a non-cyclic permutation of 1, 2, 3 0 otherwise (7) The alternating unit tensor is positive when the indices assume any clockwise cyclical progression, and it is negative when the indices assume any anticlockwise cyclic order, as shown in the figure: Since each subscript can only take on values 1, 2 or 3, it follows that ɛ ijk has 27 components. 1 Tullio Levi-Civita, 29 March 1873 Padua - 29 December 1941 Rome, was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity. Vectors & Tensors, E. Nobile March 5, / 36

15 Permutation Symbol - cont. From the definition of ɛ ijk, it follows that and all other ɛ ijk = 0, for example These results imply the following identities ɛ 123 = ɛ 231 = ɛ 312 = +1 (8) ɛ 132 = ɛ 321 = ɛ 213 = 1 (9) ɛ 223 = ɛ 233 = ɛ 311 = 0 (10) ɛ ijk = ɛ jki = ɛ kij (11) That is, a cyclic permutation of the indices on the permutation symbol does not alter its value. Also ɛ ijk = ɛ kji, ɛ ijk = ɛ jik, ɛ ijk = ɛ ikj (12) that is, interchange of any two indices on the permutation symbol introduces a minus sign. This corresponds to an anticyclic permutation of indices. Vectors & Tensors, E. Nobile March 5, / 36 Permutation Symbol and vector product With the introduction of the permutation symbol ɛ ijk it can be shown that the ith component of the vector product of two vectors a and b (or a i and b i ) is (a b) i = ɛ ijk a j b k (13) The right-hand side of (13) involves two repeated indices and therefore a double summation. 1 Firstly summation over the index j: ɛ ijk a j b k = ɛ i1k a 1 b k + ɛ i2k a 2 b k + ɛ i3k a 3 b k 2 Summing over the index k (and using the result in (10) that ɛ ijk is zero if it contains two identical indices), ɛ ijk = ɛ i11 a 1 b 1 + ɛ i12 a 1 b 2 + ɛ i13 a 1 b 3 + ɛ i21 a 2 b 1 + ɛ i22 a 2 b 2 + ɛ i23 a 2 b 3 + ɛ i31 a 3 b 1 + ɛ i32 a 3 b 2 + ɛ i33 a 3 b 3 Vectors & Tensors, E. Nobile March 5, / 36

16 Permutation Symbol and vector product - cont. Therefore if i takes the value 1: Similarly, if i = 2 and if i = 3 ɛ 1jk a j b k = ɛ 123 a 2 b 3 + ɛ 132 a 3 b 2 = a 2 b 3 a 3 b 2 ɛ 2jk a j b k = ɛ 213 a 1 b 3 + ɛ 231 a 3 b 1 = a 3 b 1 a 1 b 3 ɛ 3jk a j b k = ɛ 312 a 1 b 2 + ɛ 321 a 2 b 1 = a 1 b 2 a 2 b 1 But these three terms - just replace 1, 2, 3 with x, y, z - are the components of a b, and thus (13) is verified. Vectors & Tensors, E. Nobile March 5, / 36 The ɛ δ identity There exists a basic identity (it is proved using the properties of determinants) connecting the ɛ symbol and the δ symbol. It is ɛ ijk ɛ ipr = δ jp δ kr δ jr δ kp (14) It has numerous applications when developing vector identities. For example, it is possible to re-prove, using the index approach, the identity (1), namely a (b c) = (a c) b (a b) c Now the ith component of the left-hand side is, using (13) twice ɛ ijk a j (b c) k = ɛ ijk ɛ klm a j b l c m After a cyclic permutation on the first ɛ symbol on the right-hand side, it results ɛ kij ɛ klm a j b l c m = (δ il δ jm δ im δ jl ) a j b l c m = a j b i c j a j b j c i = (a c) b i (a b) c i = {(a c) b (a b) c} i which proves the result. Vectors & Tensors, E. Nobile March 5, / 36

17 Total differential of a function and gradient Consider the total differential of a function of three variables, φ(x 1, x 2, x 3 ). We have In tensor notation, this is replaced by dφ = φ x 1 dx 1 + φ x 2 dx 2 + φ x 3 dx 3 (15) dφ = φ x i dx i (16) Equation (16) can be thought of as the dot product of the gradient of φ, namely φ, and the differential vector dx = ( dx 1, dx 2, dx 3 ). Thus, the ith component of φ, which we denote as ( φ) i, is given by Gradient ( φ) i = φ x i = φ,i (17) where a comma followed by an index is tensor notation for differentiation with respect to x i. Vectors & Tensors, E. Nobile March 5, / 36 Divergence and curl The divergence of a vector u is given by Divergence where again differentiation with respect to x i is denoted by, i. The curl of a vector u is Curl u = u i x i = u i,i (18) ( u) = ɛ ijk u k x j = ɛ ijk u k,j (19) Vectors & Tensors, E. Nobile March 5, / 36