Appendix I Elements of Tensor Calculus Key Word Appendix 1. Elements of Tensor Calculus 695 In Brief Given a veetor spaee E and its dual spaee E*, t4e not ion of tensor arises when we study the multilinear forms on the spaee F defined as the order n Cartesian produet of E and E* (Seet. 1). One obvious way of produeing sueh forms eonsists in eonsidering those forms whose values are given by the produet of the values taken by n linear forms on E or E*. An n-linear form on F obtained in this way is ealled a product tensor of rank n, and it is written by inserting the symbol @ between eaeh of the forms on E or E* that make it up, these being themselves elements of E* or E, respeetively (Seet. 2). This eonstruetion does not generate the whole spaee of n-linear forms on F. However, it ean be used to produee a basis for the veetor spaee of n-linear forms on F, for example, given a basis of E and the dual basis for E*. All the forms we seek, known as tensors (of rank n) on F, are generated by linear eombination from sueh a basis (Seet. 3). The eorresponding veetor spaee is identified with a tensor produet of order n of E and E* . Two fundamental operations are defined on the tensors. The first is the tensor product, denoted by @. For two tensors of rank p and q, defined on spaees Fp and Fq , their tensor produet is a tensor of rank (p + q) on the product spaee F of Fp and Fq (or the other way round). This generalises the method for eonstrueting produet tensors from elements of E* and E (Seet. 2). The seeond is the contraction. This allows us, in eertain eireumstanees, to obtain tensors of rank (n - 2), (n - 4), and so forth, on spaees Fn - 2 , ... , from a tensor of rank n on F (Seet. 3). These operations ean be eombined. We obtain then the contracted product of two tensors, of great importanee in meehanics (Seet. 4). We know that for a spaee E with Euelidean strueture, there is a eanon­ ieal (natural) isomorphism whieh allows us to identify E with its dual E*, defined by substituting the sealar produet in E for the duality produet. This isomorphism ean also be used to show that the Cartesian produet spaees of E and E* of arbitrary order n are themselves isomorphie. It follows that the tensor spaees defined on these produets are also isomorphie. Henee, we identify a linear form on E, element of E*, with its assoeiated veetor, ele­ ment of E, using the eanonieal isomorphism, thereby introdueing the not ion of Euelidean veetor. In the same way, we shall identify the 2n tensors of rank 696 Appendix I. Elements of Tensor Calculus n which correspond to one another by the isomorphism with that tensor amongst them which is an element of the tensor product of rank n of E with itself: this gives us the corresponding Euclidean tensor. The two fundamental operations introduced earlier can be carried over in a consistent way to the Euclidean tensors. Contraction is always possible in this case; the associated rules are simplified and can all be expressed in terms of the scalar product on E (Sect. 5). One important application of tensor calculus is associated with the fact that we can generalise the ideas of gradient and divergence to higher orders. For a field of rank n tensors, defined on an affine space for which E is the associated vector space, the gradient at a point is the tensor of rank (n + 1) whose contracted product with an infinitesimal vector (differential element) in E gives the corresponding differential increment in the tensor field at this point. The divergence is obtained by contracting the gradient tensor. It is therefore a tensor of rank (n - 1). The divergence theorem transforming a fiux type surface integral into a volume integral is then extended to tensors of any rank (Sect. 6). Appendix I. Elements of Tensor Calculus 697 Main Notation Notation Meaning First cited (, ) Duality product (1.2) 0 Tensor product (2.1) sj, Kronecker symbol (2.6) T == Ti/§.i0e*j 0§.k Tensor (3.3) det Determinant (3.16) tr Trace (3.17) Transposition symbol (3.18) 8 Contracted product on the last index of (4.3) the tensor before 8 and the first index of the tensor coming after it 0 Doubly contracted product on the (4.14) two indices next to 0, and on the two indices next to those G Metrie tensor (5.1) Scalar product (5.2) Contraeted produet for Euelidean (5.32) tensors: the same rule as for 8 coneerning indices Doubly contracted product for (5.36) Euclidean tensors: same rule as for o eoneerning indices T.. Euelidean tensor Seet. 5.8 t. Matrix for T.. in an orlhonormal basis Seet. 5.9 DllC Derivative along veetor Y1. (6.3) \7 Gradient (6.4) div Divergenee (6.8) 698 Appendix 1. Elements of Tensor Calculus 1 Tensors on a Veetor Spaee ..................................... 699 1.1 Definition ............................................. 699 1.2 First Rank Tensors ..................................... 700 1.3 Seeond Rank Tensors ................................... 700 2 Tensor Produet of Tensors ..................................... 701 2.1 Definition ............................................. 701 2.2 Examples ............................................. 701 2.3 Produet Tensors ....................................... 702 3 Tensor Components ........................................... 703 3.1 Definition ............................................. 703 3.2 Change of Basis ........................................ 704 3.3 Mixed Seeond Rank Tensors ............................. 705 3.4 Twiee Contravariant or Twiee Covariant Seeond Rank Tensors 707 3.5 Components of a Tensor Product ......................... 708 4 Contraction .................................................. 708 4.1 Definition of the Contraction of a Tensor .................. 708 4.2 Contraeted Multiplieation ............................... 709 4.3 Doubly Contraeted Produet of Two Tensors ............... 711 4.4 Total Contraction of a Tensor Produet .................... 713 4.5 Defining Tensors by Duality ............................. 713 4.6 Invariants of a Mixed Seeond Rank Tensor ................ 714 5 Tensors on a Euclidean Vector Spaee ............................ 714 5.1 Definition of a Euelidean Spaee .......................... 714 5.2 Applieation: Deformation in a Linear Mapping ............. 715 5.3 Isomorphism Between E and E* ......................... 715 5.4 Covariant Form of Veetors in E .......................... 717 5.5 First Rank Euelidean Tensors and the Contracted Product .. 718 5.6 Seeond Rank Euclidean Tensors of Simple Produet Form and their Contraeted Produets ..... 719 5.7 Second Rank Euelidean Tensors .......................... 721 5.8 Rank n Euelidean Tensors ............................... 726 5.9 Choice of Orthonormal Basis in E . ....................... 726 5.10 Prineipal Axes and Principal Values of a Real Symmetrie Seeond Rank Euclidean Tensor ........ 727 6 Tensor Fields ................................................. 729 6.1 Definition ............................................. 729 6.2 Derivative and Gradient of a Tensor Field ................. 729 6.3 Divergenee of a Tensor Field ............................. 731 6.4 Curvilinear Coordinates ................................. 732 Summary of Main Formulas ........................................ 737 1. Tensors on a Vector Space 699 Elements of Tensor Calculus Without being too concerned with the mathematical formalism, which is available in many other works, the aim here is to provide the reader with sufficient basic knowledge to be able to use the tensor calculus in the context of 3-dimensional continuum mechanics as presented earlier in the book. This introduction to tensor calculus is organised into three parts. The first is de­ voted to the definition of tensors on a vector space, presentation of their basic properties and discussion of the main operations of tensor calculus (Sects. 1 to 4). The second part deals with Euclidean tensors (Sect. 5), and the third tackles the quest ion of tensor fields and their derivatives. Concerning immediate applications to the subject of the present book, the second and third parts (for Euclidean tensors) will be the most relevant. For this reason, the summary of the main formulas given at the end of this appendix only presents results referring to Euclidean tensors. However, it seems preferable to provide an initial discussion that brings out the role of duality, before introducing a Euclidean structure. 1. Tensors on a Vector Space 1.1 Definition Let E denote a vector space of finite dimension n (over ~ or C) and let E* be the dual of E. Then a p times contravariant and q times covariant tensor is any multilinear form T defined on (E*)P x (E)q. Denote by u*(i) p arbitrary vectors in E* i = 1, ... ,p , 12.(j) q arbitrary vectors in E j = 1, ... ,q . Such a multilinear form associates the following scalar with the vector argu­ ments u*(i) and 1I.(j)' taken in this order: T(u*(l), ... ,u*(p) ,12.(1)' ... ,12.(q)) . (1.1) The sum (p + q) is called the rank of the tensor. The pair of numbers (p, q) is called its type. The order in which the vector arguments occur in T must be specified in the definition of the form. In this presentation, we have chosen to order the arguments by first taking the vectors in E* and then taking those in E. It is clear that the set of all tensors of given type (p, q), and corresponding to the same order of vector arguments, can be supplied with a vector space structure. 700 Appendix I. Elements of Tensor Calculus As an example, let us examine the first and second rank tensors, these being widely used in continuum mechanics. Notation: To begin with, let (u*, '1)) = ÜL, u*) (1.2) denote the duality product between a vector u* in E* and a vector :l!. in E. 1.2 First Rank Tensors First Rank Contravariant Tensor From the definition, this is a linear form T on E*, identified classically with a vector 'L of E by means of the duality product.