APPENDIX B Tensors on a vector space In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the “modern”, geometrical view on tensors, which defines them by their action on vectors or one-forms, i.e. in a coordinate-independent way (Sec. B.1), in contrast to the “old” definition based on their behavior under basis transformations (Sec. B.2). The reader is assumed to already possess enough knowledge on linear algebra to know what are vectors, linear (in)dependence, (multi)linearity, matrices. Similarly, the notions of group, field, application/function/mapping. are used without further mention. In the remainder of these lecture notes, we actually consider tensors on real vector spaces, i.e. for which the underlying base field K of scalars is the set R of real numbers; here we remain more general. Einstein’s summation convention is used throughout. B.1 Vectors, one-forms and tensors B.1.1 Vectors . are by definition the elements ~c of a vector space V , i.e. of a set with 1) a binary operation (“addition”) with which it is an Abelian group, and 2) a multiplication with “scalars”—elements of a base field K—which is associative, has an identity element, and is distributive with respect to both additions on V and on K. Introducing a basis B = f~eig, i.e. a family of linearly independent vectors that span the whole space V , one associates to each vector ~c its uniquely defined components fcig, elements of the base field K, such that i ~c = c ~ei: (B.1) If the number of vectors of a basis is finite—in which case this holds for all bases— and equal to some integer D—which is the same for all bases—, the space V is said to be finite-dimensional and D is its dimension (over K): D = dim V . We shall assume that this is the case in the remainder of this Section. B.1.2 One-forms . on a vector space V are the linear applications, hereafter denoted as h, from V into the base field of scalars K. e The set of 1-forms on V , equipped with the “natural” addition and scalar multiplication, is itself a vector space over the field K, denoted by V ∗ and said to be dual to V . ∗ ∗ If V is finite-dimensional, so is V , with dim V = dim V . Given a basis B = f~eig in V , one can then construct its dual basis B∗ = fjg in V ∗ such that e j j (~ei) = δi ; (B.2) j e where δi denotes the usual Kronecker delta symbol. The components of a 1-form h on a given basis will be denoted as fhjg: j e h = hj : (B.3) e e B.1 Vectors, one-forms and tensors 53 Remarks: ∗ The choice of notations, in particular the position of indices, is not innocent! Thus, if fjg denotes the dual base to f~eig, the reader can trivially check that e i i c = (~c) and hj = h(~ej): (B.4) e e ∗ In the “old” language, the vectors of V resp. the 1-forms of V ∗ were designated as “contravariant vectors” resp. “covariant vectors” or “covectors”, and their coordinates as “contravariant” resp. ”co- variant” coordinates. The latter two, applying to the components, remain useful short denominations, especially when applied to tensors (see below). Yet in truth they are not different components of a same mathemat- ical quantity, but components of different objects between which a “natural” correspondence was introduced, in particular by using a metric tensor as in § B.1.4. B.1.3 Tensors :::::::B.1.3 a :::::::::::::::::::::::::::Definition and first results Let V be a vector space with base field K, and m, n denote two nonnegative integers. The multilinear applications of m one-forms—elements of V ∗—and n vectors—elements of V —into m K are referred to as the tensors of type n on V , where linearity should hold with respect to every argument. The integer m + n is the order (or often, but improperly, rank) of the tensor. Already known objects arise as special cases of this definition when either m or n is zero: 0 K • the 0 -tensors are simply the scalars of the base field ; 1 (19) • the 0 -tensors coincide with vectors; 0 0 • the 1 -tensors are the one-forms. More generally, the n -tensors are also known as (multi- linear) n-forms. 2 • Eventually, 0 -tensors are sometimes called “bivectors” or “dyadics”. Tensors will generically be denoted as T, irrespective of their rank, unless the latter is 0 or 1. A tensor may be symmetric or antisymmetric under the exchange of two of its arguments, either both vectors or both 1-forms. Generalizing, it may be totally symmetric—as e.g. the metric tensor we shall encounter below—, or antisymmetric. An instance of the latter case is the determinant, which is the only (up to a multiplicative factor) totally antisymmetric D-form on a vector space of dimension D. m ∗ m ∗ n 0 0 Remark: Consider a n -tensor T :(V ) ×(V ) ! K, and let m ≤ m, n ≤ n be two nonnegative 0 0 integers. For every m -uplet of one-forms fhig and n -uplet of vectors f~cjg—and corresponding multiplets of argument positions, although heree we take for simplicity the first ones—the object T h1; : : : ; hm0 ; · ;:::; · ;~c1; : : : ;~cn0 ; · ;:::; · ; e e where the dots denote “empty” arguments, can be applied to m − m0 one-forms and n − n0 vectors 0 0 to yield a scalar. That is, the tensor T induces a multilinear application(20) from (V ∗)m × (V ∗)n m−m0 into the set of n−n0 -tensors. 1 For example, the 1 -tensors are in natural correspondence with the linear applications from V into V , i.e. in turn with the square matrices of order dim V . (19)More accurately, they are the elements of the double dual of V , which is always homomorphic to V . (20)Rather, the number of such applications is the number of independent—under consideration of possible symmetries—combinations of m0 resp. n0 one-form resp. vector arguments. 54 Tensors on a vector space :::::::B.1.3 b :::::::::::::::::::::::Operations on tensors The tensors of a given type, with the addition and scalar multiplication inherited from V ; form a vector space on K. Besides these natural addition and multiplication, one defines two further operations on tensors, the outer product or tensor product—which increases the rank—and the contraction, which decreases the rank. 0 m m0 0 Consider two tensors T and T , of respective types n and n0 . Their outer product T ⊗T is m+m0 0 0 a tensor of type n+n0 satisfying for every (m + m )-uplet (h1; : : : ; hm; : : : ; hm+m ) of 1-forms and 0 every (n + n )-uplet (~c1; : : : ;~cn; : : : ;~cn+n0 ) of vectors the identitye e e 0 T ⊗ T h1; : : : ; hm+m0 ;~c1; : : : ;~cn+n0 = e e 0 T h1; : : : ; hm;~c1; : : : ;~cn T hm+1; : : : ; hm+m0 ;~cn+1; : : : ;~cn+n0 : e e e e For instance, the outer product of two 1-forms h, h0 is a 2-form h ⊗ h0 such that for every pair of vectors (~c;~c 0), h ⊗ h0(~c;~c 0) = h(~c) h0(~c 0). In turn,e e the outer producte e of two vectors ~c, ~c 0 is a 2 0 0 0 0 0 0 0 -tensor ~c ⊗ ~c suche thate for everye paire of 1-forms (h; h ), ~c ⊗ ~c (h; h ) = h(~c) h (~c ). m Tensors of type n that can be written as outere productse ofeme vectorse ande n one-forms are sometimes called simple tensors. m Let T be a n -tensor, where both m and n are non-zero. To define the contraction over its j-th one-form and k-th vector arguments, the easiest—apart from introducing the tensor components—is to write T as a sum of simple tensors. By applying in each of the summand the k-th one-form to m−1 the j-th vector, which gives a number, one obtains a sum of simple tensors of type n−1 , which is the result of the contraction operation. Examples of contractions will be given after the metric tensor has been introduced. :::::::B.1.3 c ::::::::::::::::::::Tensor coordinates j ∗ Let f~eig resp. f g denote bases on a vector space V of dimension D resp. on its dual V —in principle, they neede not be dual to each other, although using dual bases is what is implicitly always done in practice—and m, n be two nonnegative integers. m+n j1 jn The D simple tensors f~ei1 ⊗ · · · ⊗~eim ⊗ ⊗ · · · ⊗ g, where each ik or jk runs from 1 to D, m form a basis of the tensors of type n . The componentse e of a tensor T on this basis will be denoted i1:::im as fT j1:::jn g: i1:::im j1 jn T = T j1:::jn ~ei1 ⊗ · · · ⊗~eim ⊗ ⊗ · · · ⊗ ; (B.5a) e e where i1:::im i1 im T j1:::jn = T( ; : : : ; ;~ej1 ; : : : ;~ejn ): (B.5b) e e The possible symmetry or antisymmetry of a tensor with respect to the exchange of two of its arguments translates into the corresponding symmetry or antisymmetry of the components when exchanging the respective indices. In turn, the contraction of T over its j-th one-form and k-th vector arguments yields the tensor :::ij−1;`;ij+1;::: with components T :::jk−1;`;jk+1;:::, with summation over the repeated index `.