Gill Williamson 15Chap11

543 C H A P T E R 11 Multilinear Mappings and Tensors In this chapter we generalize our earlier discussion of bilinear forms, which leads in a natural manner to the concepts of tensors and tensor products. While we are aware that our approach is not the most general possible (which is to say it is not the most abstract), we feel that it is more intuitive, and hence the best way to approach the subject for the first time. In fact, our treatment is essen tially all that is ever needed by physicists, e ngineers and applied mathemati cians . More general treatments are discussed in advanced courses on abstract algebra. The basic idea is as follows. Given a vector space V with basis {eá}, we defined the dual space V* (with basis {ø i}) as the space of linea r functionals on V. In other words, if ƒ = Íáƒáø i ∞ V* and v = Íév jeé ∞ V, then !(v)=Ó!,!vÔ=Ó"i!i#i,!"jvjejÔ="i,j!ivj$ij="i!ivi!!. Next we defined the space B(V) of all bilinear forms on V (i.e., bilinear map- pings on V ª V), and we showed (Theorem 9.10) that B(V) has a basis given by {f ij = ø i · ø j} where fij(u, v) = øi · ø j(u, v) = ø i(u)ø j(v) = u ivj . MULTILINEAR MAPPINGS AND TENSORS 544 It is this definition of the f ij that we will now generalize to include linear func- tionals on spaces such as, for example, V* ª V* ª V* ª V ª V. 11.1 DEFINITIONS Let V be a finite -dimensional vector space ove r F, and let V r denote the r-fold Cartesian product V ª V ª ~ ~ ~ ª V. In other words, an element of V r is an r- tuple (vè, . . . , v r) where each vá ∞ V. If W is another vector space over F, then a mapping T: V r ‘ W is said to be multilinear if T(vè, . . . , v r) is linear in each variable. That is, T is multilinear if for each i = 1, . . . , r we have T(vè, . . . , avá + bvæá , . . . , v r) = aT(vè, . . . , vá, . . . , v r) + bT(vè, . . . , væá, . . . , v r) for all vá, væá ∞ V and a, b ∞ F. In the particular case that W = F, the mapping T is variously called an r-linear form on V, or a multilinear form of degree r on V, or an r-tensor on V. The set of all r-tensors on V will be denoted by Tr (V). (It is also possible to discuss multilinear mappings that take their values in W rather than in F. See Section 11.5 .) As might be expected, we define addition and scalar multiplication on Tr (V) by (S+T)(v1,!!…!,!vr)=S(v1,!!…!,!vr)+T(v1,!!…!,!vr)(aT)(v1,!!…!,!vr)=aT(v1,!!…!,!vr) for all S, T ∞ Tr (V) and a ∞ F. It should be clear that S + T and aT are both r- tensors . With these operations, Tr (V) becomes a vector space over F. Note that the particular case of r = 1 yields T1 (V) = V*, i.e., the dual space of V, and if r = 2, then we obtain a bilinear form on V. Although this definition takes care of most of what we will need in this chapter, it is worth going through a more general (but not really more difficult) definition as follows. The basic idea is that a tensor is a scalar - valued multi linear function with variables in both V and V*. Note also that by Theorem 9.4, the space of linear functions on V* is V** which we view as simply V. For example, a tensor could be a function on the space V* ª V ª V. By convention , we will alw ays write all V* variables before all V variables, so that, for example, a tensor on V ª V* ª V will be replaced by a tensor on V* ª V ª V. (However, not all authors adhere to this convention, so the reader should be very careful when reading the literatur e.) 11.1 DEFINITIONS 545 Without further ado, we define a tensor T on V to be a multilinear map on V*s ª V r: T:!V!s"Vr=V!"!"V!s copies"#$%$"V"!"Vr copies"#$%$#F where r is called the covariant order and s is called the contravariant order of T. We shall say that a tensor of covariant order r and contravariant order s is of type (or rank ) (rÍ). If we denote the set of all tensors of type ( rÍ) by Tr Í(V), then defining addition and scalar multiplication exactly as above, we see that Tr Í(V) forms a vector space over F. A tensor of type ( 0º) is defined to be a scalar, and hence T0 º(V) = F. A tensor of type ( 0¡) is called a contravariant vector , and a tensor of type ( 1º) is called a covariant vector (or simply a covector ). In order to distinguish between these types of vectors, we denote the basis vectors for V by a subscript (e.g., eá), and the basis vectors for V* by a supe rscript (e.g., ø j). Furthermore, we will generally leave off the V and simply write Tr or Tr Í. At this point we are virtually forced to introduce the so-called Einstein summation convention . This convention says that we are to sum over repeated indices in any vector or tensor expression where one index is a super script and one is a subscript. Because of this, we write the vector com- ponents with indices in the opposite position from that of the basis vectors. This is why we have been writing v = Íáv ieá ∞ V and ƒ = Íéƒéø j ∞ V*. Thus we now simply write v = v ieá and ƒ = ƒéø j where the summation is to be understood. Generally the limits of the sum will be clear. However, we will revert to the more complete notation if there is any possibility of ambiguit y. It is also worth emphasizing the trivial fact that the indices summed over are just “dummy indices. ” In other words, we have v iei = v jej and so on. Throughout this chapter we will be relabelling indices in this manner without further notice, and we wil l assume that the reader understands what we are doing. Suppose T ∞ Tr, and let {eè, . . . , eñ} be a basis for V. For each i = 1, . . . , r we define a vector vá = eéa ji where, as usual, a ji ∞ F is just the j th component of the vector vá. (Note that here the subscript i is not a tensor index.) Using the multi linearity of T we see that T(vè, . . . , v r) = T(e jèajè1, . . . , e j‹aj‹r) = a jè1 ~ ~ ~ a j‹r T(e jè , . . . , e j‹) . The n r scalars T(e jè , . . . , e j‹) are called the components of T relative to the basis {eá}, and are den oted by T jè ~ ~ ~ j‹ . This terminology implies that there exists a basis for Tr such that the T jè ~ ~ ~ j‹ are just the components of T with MULTILINEAR MAPPINGS AND TENSORS 546 respect to this basis. We now construct this basis, which will prove that Tr is of dimension n r. (We will show f ormally in Section 11.10 that the Kronecker symbols ∂ij are in fact the components of a tensor, and that these components are the same in any coordinate system. However, for all practical purposes we continue to use the ∂ij simply as a notational device, a nd hence we place no importance on the position of the indices, i.e., ∂ij = ∂ji etc.) For each collection {iè, . . . , i r} (where 1 ¯ iÉ ¯ n), we define the tensor Øiè ~ ~ ~ i‹ (not simply the components of a tensor Ø) to be that element of Tr whose value s on the basis {eá} for V are given by Øiè ~ ~ ~ i‹ (ejè , . . . , e j‹) = ∂ièjè ~ ~ ~ ∂i‹j‹ and whose values on an arbitrary collection {vè, . . . , v r} of vectors are given by multilinearity as !i1!ir(v1,!!…!,!vr)=!i1!…!ir(ej1aj11,!!…!,!ejrajrr)=aj11!ajrr!i1!ir(ej1,!!…!,!ejr)=aj11!ajrr"i1j1!"irjr=ai11!airr!!. That this does indeed define a tensor is guaranteed by this last equation which shows that each Ø iè ~ ~ ~ i‹ is in fact linear in each variable (since vè + væè = (ajè1 + aæ jè1)ejè etc.). To prove that the n r tensors Ø iè ~ ~ ~ i‹ form a basis for Tr, we must show that they linearly in dependent and span Tr. Suppose that å iè ~ ~ ~ i‹ Øiè ~ ~ ~ i‹ = 0 where each å iè ~ ~ ~ i‹ ∞ F. From the definition of Ø iè ~ ~ ~ i‹, we see that applying this to any r-tuple (ejè , . . . , e j‹) of basis vectors yields å jè ~ ~ ~ j‹ = 0. Since this is true for every such r-tuple , it follows that å iè ~ ~ ~ i‹ = 0 for every r-tuple of indices (iè , . . . , i r), and hence the Øiè ~ ~ ~ i‹ are linearly independent. Now let T iè ~ ~ ~ i‹ = T(e iè , . . . , e i‹) and consider the tensor Tiè ~ ~ ~ i‹ Øiè ~ ~ ~ i‹ in Tr. Using the definition of Ø iè ~ ~ ~ i‹ , we see that both T iè ~ ~ ~ i‹ Øiè ~ ~ ~ i‹ and T yield the same result when applied to any r-tuple (ejè , . . . , e j‹) of basis vectors, and hence they must be equal as multilinear functions on V r. This shows that {Ø iè ~ ~ ~ i‹ } spans Tr. 11.1 DEFINITIONS 547 While we have treated only the space Tr, it is not any more dif ficult to treat the general space Tr Í. Thus, if {eá} is a basis for V, {ø j} is a basis for V* and T ∞ Tr Í, we define the components of T (relative to the given bases) by Tiè ~ ~ ~ i› jè ~ ~ ~ j‹ = T(ø iè , . . . , ø i›, ejè , . . . , e j‹ ) . Defining the n r+s analogous tensors Ø i!Ô⁄`Ÿ`Ÿ`Ÿi$Ô‹ , it is easy to mimic the above procedure and hence prove the following result. Theorem 11.1 The set Tr Í of all tensors of type ( rÍ) on V forms a vector space of dimension n r+s. Proof This is Exercise 11.1.1. ˙ Since a tensor T ∞ Tr Í is a function on V* s ª V r, it would be nice if we could write a basis (e.g., Ø i!Ô⁄`Ÿ`Ÿ`Ÿi$Ô‹ ) for Tr Í in terms of the bases {eá} for V and {øj} for V*. We now show that this is easy to accomplish by defining a product on Tr Í, called the tensor product. The reader is cautioned not to be intimidated by the notational complexities, since the concepts involved are really quite simple. Suppose that S!Tr1s1 and T!Tr2s2 . Let uè, . . . , u rè, vè, . . . , v rì be vectors in V, and å 1, . . . , åsè, ∫1, . . . , ∫sì be covectors in V*. Note that the product S(å 1, . . . , å sè, uè, . . . , u rè) T(∫1, . . . , ∫sì, vè, . . . , v rì) is linear in each of its rè + sè + rì + sì variables. Hence we define the tensor product S · T ∞ Tr1+r2s1+s2 (read “S tensor T ”) by (S · T)(å 1, . . . , å sè, ∫1, . . . , ∫sì, uè, . . . , u rè, vè, . . . , v rì) = S(å 1, . . . , å sè, uè, . . . , u rè ) T(∫1, . . . , ∫sì, vè, . . . , v rì ) . It is easily shown that the tensor product i s both associative and distribu - tive (i.e., bilinear in both factors). In other words, for any scalar a ∞ F and tensors R, S and T such that the following formulas make sense, we have (R!S)!T=R!(S!T)R!(S+T)=R!S+R!T(R+S)!T=R!T+S!T(aS)!T=S!(aT)=a(S!T) MULTILINEAR MAPPINGS AND TENSORS 548 (see Exercise 11.1.2). Because of the assoc iativity property (which is a conse - quence of associativity in F), we will drop the parentheses in expressions such as the top equation and simply write R · S · T. This clearly extends to any finite product of tensors. It is important to note, however, tha t the tensor product is most certainly not commutative. Now let {eè, . . . , eñ} be a basis for V, and let {ø j} be its dual basis. We claim that the set {ø jè · ~ ~ ~ · ø j‹} of tensor products where 1 ¯ jÉ ¯ n forms a basis for the space Tr of covariant te nsors. To see this, we note that from the definitions of tensor product and dual space, we have øjè · ~ ~ ~ · ø j‹(eiè , . . . , e i‹) = ø jè(eiè) ~ ~ ~ ø j‹(ei‹) = ∂jèiè ~ ~ ~ ∂j‹i‹ so that ø jè · ~ ~ ~ · ø j‹ and Ø jè ~ ~ ~ j‹ take the same values on the r-tuples (eiè, . . . , e i‹), and hence they must be equal as multilinear functions on V r. Since we showed above that {Ø jè ~ ~ ~ j‹} forms a basis for Tr , we have proved that {ø jè · ~ ~ ~ · ø j‹} also forms a basis for Tr . The method of the previous par agraph is readily extended to the space Tr Í. We must recall however, that we are treating V** and V as the same space. If {eá} is a basis for V, then the dual basis {ø j} for V* was defined by ø j(eá) = Óøj, eá Ô = ∂já. Similarly, given a basis {ø j} for V*, we define the basis {eá} for V** = V by eá(ø j) = ø j(eá) = ∂já. In fact, using tensor products, it is now easy to repeat Theorem 11.1 in its most useful form. Note also that the next theorem shows that a tens or is determined by its values on the bases {eá} and {øj}. Theorem 11.2 Let V have basis {eè, . . . , eñ}, and let V* have the corre - sponding dual basis {ø 1, . . . , ø n}. Then a basis for Tr Í is given by the collec - tion {eiè · ~ ~ ~ · e i› · ø jè · ~ ~ ~ · ø j‹} where 1 ¯ jè, . . . , j r, iè, . . . , i s ¯ n, and hence dim Tr Í = n r+s. Proof In view of Theorem 11.1, all that is needed is to show that eiè · ~ ~ ~ · e i› · ø jè · ~ ~ ~ · ø j‹ = Ø i!Ô⁄`Ÿ`Ÿ`Ÿi$Ô‹ . The details are left to the reader (see Exercise 11.1.1). ˙ Since the components of a tensor T are defined with respect to a particular basis (and dual basis), we might ask about the relationship between the com- 11.1 DEFINITIONS 549 ponents of T relative to two different bases. Using the multilinearity of tensors, this is a simple problem to solve. First, let {eá} be a basis for V and let {ø j} be its dual basis. If {eõá} is another basis for V, then there exists a nonsingular transition matrix A = (a ji) such that eõi = e jaji . (1) (We emphasize that a já is only a matrix, not a tensor. Note also that our defini - tion of the matrix of a linear transformation given in Section 5.3 shows that a já is the element of A in the j th row and i th column.) Using Óøi, eéÔ = ∂ié, we have Óø i, eõÉÔ = Óø i, eéa jÉÔ = a jÉÓøi, eéÔ = a jÉ∂ié = a iÉ . Let us denote the inverse of the matrix A = (a ié) by Aî = B = (b ié). In other words, a iébjÉ = ∂iÉ and b iéajÉ = ∂iÉ. Multiplying Óø i, eõÉÔ = a iÉ by b já and summing on i yields Ób jáøi, eõÉÔ = b jáaiÉ = ∂jÉ . But the basis {øù i} dual to {eõá} also must satisfy Óøù j, eõkÔ = ∂jÉ, and hence comparing this with the previous equation shows that the dual basis vectors transform as øùj = b jiøi (2) The reader sh ould compare this carefully with (1). We say that the dual basis vectors transform oppositely (i.e., use the inverse transformation matrix) to the basis vectors. It is also worth emphasizing that if the nonsingular transi - tion matrix from the basis {eá} to the basis {eõá} is given by A, then (according to the same convention given in Section 5.4) the corresponding nonsingular transition matrix from the basis {ø i} to the basis {øù i} is given by B T = (Aî) T. We leave it to the reader to write out equations (1) and (2) in matrix notation to show that this is true (see Exercise 11.1.3). We now return to the question of the relationship between the components of a tensor in two different bases. For definiteness, we will consider a tensor T ∞ T1 ™. The analogous result for an arbitrary tensor in Tr Í will be quite obvious. Let {eá} and {ø j} be a basis and dual basis for V and V* respective - ly. Now consider another pair of bases {eõá} and {øù j} where eõá = eéa já and øù i = biéøj. Then we have T ij k = T(ø i, øj, ek) as well as Tä pq r = T(øù p, øùq, eõr), and there fore MULTILINEAR MAPPINGS AND TENSORS 550 Täp q r = T(øù p, øùq, eõr) = b pi bqj akr T(ø i, øj, ek) = b pi bqj akr Tijk . This is the classical law of transformation of the components of a tens or of type ( 1™). It should be kept in mind that (a ié) and (b ié) are inverse matrices to each other. (In fact, this equation is frequently taken as the definition of a tensor (at least in older texts). In other words, according to this approach, any quantit y with this transformation property is defined to be a tensor.) In particular, the components v i of a vector v = v i eá transform as või = b iévj while the components åá of a covector å = åáø i transform as åùá = åéa já . We leave it to the reader to verify that these transformation laws lead to the self-consistent formulas v = v ieá = võ j eõé and å = åáø i = åùéøù j as we should expect (see Exercise 11.1.4). We point out that these transformation laws are the origin of the terms “contravariant ” and “covariant. ” This is because the components of a vector transform oppositely ( “contravariant ”) to the basis vectors e i, while the com- ponents of dual vectors transform the same as ( “covariant ”) these basis vec- tors. It is also worth mentioning that many a uthors use a prime (or some other method such as a different type of letter) for distinguishing different bases. In other words, if we have a basis {eá} and we wish to transform to another basis which we denote by {e iæ}, then this is accomplished by a tran sformation matrix (a ijæ) so that e iæ = eéa jiæ. In this case, we would write ø iæ = a iæéøj where (aiæé) is the inverse of (a ijæ). In this notation, the transformation law for the tensor T used above would be written as Tpæqæræ = b pæibqæjakræTijk . Note that specifying the components of a tensor with respect to one coor- dinate system allows the determination of its components with respect to any other coordinate system. Because of this, we shall frequently refer to a tensor by its “generic ” components . In other words, we will refer to e.g., T ijk, as a “tensor ” and not the more accurate description as the “components of the tensor T. ” 11.1 DEFINITIONS 551 Example 11.1 For those readers who may have seen a classical treatment of tensors and have had a course in advanced c alculus, we will now show how our more modern approach agrees with the classical. If {x i} is a local coordinate system on a differentiable manifold X, then a (tangent) vector field v(x) on X is defined as the derivative function v = vi($/$x i), so that v( f ) = v i($f/$x i) for every smooth function f: X ‘ ® (and where each v i is a function of position x ∞ X, i.e., v i = v i(x)). Since every vector at x ∞ X can in this manner be written as a linear combination of the $/$x i, we see that {$/$x i} forms a basis for the tangent space at x. We now define the differential df of a function by df(v) = v(f) and thus df(v) is just the directional derivative of f in the direction of v. Note that dxi(v) = v(x i) = v j($xi/$xj) = v j∂ié = v i and hence df(v) = v i($f/$x i) = ($f/$x i)dxi(v). Since v was arbitrary, we obtain the familiar elementary formula df = ($f/$x i)dxi. Furthermore, we see that dxi($/$x j) = $x i/$xj = ∂ié so that {dx i} forms the basis dual to {$/$x i}. In summa ry then, relative to the local coordinate system {x i}, we define a basis {eá = $/$x i} for a (tangent) space V along with the dual basis {ø j = dx j} for the (cotangent) space V*. If we now go to a new coordinate system {xõ i} in the same coordinate patch, th en from calculus we obtain $/$xõ i = ($x j/$xõi)$/$x j so that the expression eõá = eéa já implies a já = $x j/$xõi. Similarly, we also have dxõi = ($xõ i/$xj)dxj so that øù i = b iéøj implies b ié = $xõ i/$xj. Note that the chain rule from calcu lus shows us that aiÉbké = ($x i/$xõk)($xõ k/$xj) = $x i/$xj = ∂ié and thus (b ié) is indeed the inverse matrix to (a ié). Using these results in the above expression for Tä pqr, we see that MULTILINEAR MAPPINGS AND TENSORS 552 Tpqr=!xp!xi!xq!xj!xk!xrTijk which is just the classical definition of th e transformation law for a tensor of type ( 1™). We also remark that in older texts, a contravariant vector is defined to have the same transformation properties as the expression dxõ i = ($xõ i/$xj)dxj, while a covariant vector is defined to have the same t ransformation properties as the expression $/$xõ i = ($x j/$xõi)$/$x j. ∆ Finally, let us define a simple classical tensor operation that is frequently quite useful. To begin with, we have seen that the result of operating on a vector v = v ieá ∞ V with a d ual vector å = åéø j ∞ V* is just Óå, vÔ = åéviÓøj, eáÔ = åév i∂já = åáv i. This is sometimes called the contraction of å with v. We leave it to the reader to show that the contraction is independent of the particular coordinate system used (see Exercise 11.1 .5). If we start with tensors of higher order, then we can perform the same sort of operation. For example, if we have S ∞ T2 ¡ with components S ijk and T ∞ T 2 with components T pq, then we can form the ( 1™) tensor with components SiéÉTjq, or a different (1™) tensor with components S iéÉTpj and so forth. This operation is also called contraction . Note that if we start with a ( 1¡) tensor T, then we can contract the components of T to obtain the scalar T iá. This is called the trace of T. Exercises 1. (a) Prove Theorem 11.1. (b) Prove Theorem 11.2. 2. Prove the four associative and distributive properties of the tensor product given in the text following Theorem 11.1. 3. If the nonsingular transition matr ix from a basis {eá} to a basis {eõá} is given by A = (a ij), show that the transition matrix from the corresponding dual bases {ø i} and {øù i} is given by (Aî) T. 4. Using the transformation matrices (a ié) and (b ié) for the bases {eá} and {eõá} and the corr esponding dual bases {ø i} and {øù i}, verify that v = v ieá = võ j eõé and å = åáø i = åùéøù j. 11.1 DEFINITIONS 553 5. If v ∞ V and å ∞ V*, show that Óå, vÔ is independent of the particular basis chosen for V. Generalize this to arbitrary tensors. 6. Let A i be a covariant vector field (i.e., A i = A i(x)) with the transformation rule Ai=!xj!xiAj!!. Show that the quantity $ jAi = $A i/$xj does not define a tensor, but that Fij = $iAj - $jAi is in fact a second -rank tensor. 11.2 SPECIAL TYPES OF TENSORS In order to obtain some of the most useful results concerning tensors, we turn our attention to the space Tr of covariant tensors on V. Generalizing our earlier definition for bilinear forms, we say that a tensor S ∞ Tr is symmet ric if for each pair (i, j) with 1 ¯ i, j ¯ r and all vá ∞ V we have S(vè, . . . , vá, . . . , vé, . . . , v r) = S(vè, . . . , vé, . . . , vá, . . . , v r) . Similarly, A ∞ Tr is said to be antis ymmetric (or skew -symmetric or alter - nating ) if A(vè, . . . , vá, . . . , vé, . . . , v r) = -A(vè, . . . , vé, . . . , vá, . . . , v r) . Note this definition implies that A(vè, . . . , v r) = 0 if any two of the vá are identical. In fact, this was the original definition of an alternating bilinear form. Furthermore, we also see that A(vè, . . . , v r) = 0 if any vá is a linear combination of the rest of the vé. In particular, this means tha t we must always have r ¯ dim V if we are to have a nonzero antisymmetric tensor of type ( rº) on V. It is easy to see that if Sè, Sì ∞ Tr are symmetric, then so is aSè + bSì where a, b ∞ F. Similarly, aAè + bAì is antisymmetric. Therefore the symmet - ric tensors form a subspace of Tr which we denote by ∑r(V), and the anti- symmetric tensors form another subspace of Tr which is denoted by „ r(V) (some authors denote this space by „ r(V*)). Elements of „ r(V) are generally called exterior r -forms , or simply r-forms . According to this terminology, the basis vectors {ø i} for V* are referred to as basis 1 -forms . Note that the only ele ment common to both of these subspaces is the zero tensor. MULTILINEAR MAPPINGS AND TENSORS 554 A partic ularly important example of an antisymmetric tensor is the determi nant function det ∞ Tn(®n) (see Theorem 4.9 and the discussion pre- ceding it). Note also that the definition of a symmetric tensor translates into the obvious requirement that (e.g., in th e particular case of T2) Sáé = Séá, while an antisym metric tensor obeys Aáé = -Aéá. These definitions can also be extended to include contravariant tensors, although we shall have little need to do so. It will be extremely convenient for us to now incorp orate the treatment of permutation groups given in Section 1.2. In terms of any permutation ß ∞ Sr, we may rephrase the above definitions as follows. We say that S ∞ Tr is sym- metric if for every collection vè, . . . , v r ∞ V and each ß ∞ Sr we have S(vè, . . . , v r) = S(v ß1 , . . . , v ßr) . Similarly, we say that A ∞ Tr is antisymmetric (or alternating ) if either A(vè, . . . , v r) = (sgn ß)A(v ß1 , . . . , v ßr) or A(v ß1 , . . . , v ßr) = ( sgn ß)A(vè, . . . , v r) where the last equation follows from the first since (sgn ß) 2 = 1. Note that even if S, T ∞ ∑r(V) are both symmetric, it need not be true that S · T be symmetric (i.e., S · T ! ∑r+r(V)). For example, if Sáé = Séá and T pq = T qp, it does not necessarily follow that SáéT pq = S ipTjq. It is also clear that if A, B ∞ „r(V), then we do not necessarily have A · B ∞ „r+r(V). Example 11.2 Suppose å ∞ „n(V), let {eè, . . . , eñ} be a basis for V, and for each i = 1, . . . , n let vá = eéa ji where a ji ∞ F. Then, using the multilinearity of å, we may write å(vè, . . . , v n) = a jè1 ~ ~ ~ a jñnå(e jè , . . . , e jñ) where the sums are over all 1 ¯ jÉ ¯ n. But å ∞ „n(V) is antisymmetric, and hence (e jè , . . . , e jñ) must be a permutation of (eè, . . . , eñ) in order that the e jÉ all be distinct (or else å(e jè , . . . , e jñ) = 0). This means that we are left with å(vè, . . . , vñ) = Í Jajè1 ~ ~ ~ a jñn å(e jè , . . . , e jñ) where Í J denotes the fact that we are summing ov er only those values of jÉ such that (jè, . . . , jñ) is a permutation of (1, . . . , n). In other words, we have 11.2 SPECIAL TYPES OF TEN SORS 555 å(vè, . . . , vñ) = Í ß´Sñ aß11 ~ ~ ~ a ßnn å(e ß1 , . . . , e ßn) . But now, by the antisymmetry of å, we see that å(e ß1 , . . . , e ßn) = (sgn ß)å(eè, . . . , eñ) and hence we are left with å(vè, . . . , vñ) = Í ß´Sñ (sgn ß) aß11 ~ ~ ~ a ßnn å(e 1 , . . . , e n) . (*) Using the definition of determinant and the fact that å(eè, . . . , eñ) is just some scalar, we finally obtain å(vè, . . . , vñ) = det(a ji) å(eè, . . . , eñ) . Referring back to Theorem 4.9, let us consider the special case where å(eè, . . . , eñ) = 1. Note that if {ø j} is a basis for V*, then øßj(vá) = ø ßj(ekaki) = a ki øßj(ek) = a ki ∂ßjk = a ßji . Using the definition of tensor product, we can therefore write (*) as det(a ji) = å(vè, . . . , vñ) = Í ß´Sñ (sgn ß)ø ß1 · ~ ~ ~ · ø ßn(vè, . . . , vñ) which implies that the determinant function is given by å = Í ß´Sñ (sgn ß)ø ß1 · ~ ~ ~ · ø ßn . In other words, if A is a matrix with columns given by vè, . . . , vñ then det A = å(vè, . . . , vñ). While we went through many detailed manipulations in arriving at these equations, we will assume from now on that the reader understand s what was done in this example , and henceforth leave out some of the intermediate steps in such calculations. ∆ At the risk of boring some readers, let us very briefly review the meaning of the binomial coefficient ( rˆ) = n!/[r!(n - r)!]. The idea is that we want to know the number of ways of picking r distinct objects out of a collection of n distinct objects. In other words, how many combinations of n things taken r at a time are there? Well, to pick r objects, we have n choices for the first, then n - 1 choices for the second, and so on down to n - (r - 1) = n - r + 1 choices for the r th . This gives us n(n - 1) ~ ~ ~ (n - r + 1) = n!/(n - r)! MULTILINEAR MAPPINGS AND TENSORS 556 as the number of ways of picking r objects out of n if we take into account the order in which the r objects are chosen. In other words, this is the number of injections INJ(r, n) (see Section 4.6). For example, to pick three numbers in order out of the set {1, 2, 3, 4}, we might choose (1, 3, 4), or we could choose (3, 1, 4). It is this kind of situation that we must take into account. But for each distinct collection of r objects, there are r! ways of arranging these, and hence we have over-counted each collection by a factor of r!. Dividing by r! then yields the desired result. If {eè, . . . , eñ} is a basis for V and T ∞ „r(V), then T is determined by its values T(e iè , . . . , e i‹) for iè < ~ ~ ~ < i r. Indeed, following the same procedure as in Example 11.2, we see that if vá = eéa ji for i = 1, . . . , r then T(vè, . . . , v r) = a iè1 ~ ~ ~ a i‹rT(eiè , . . . , e i‹) where each sum is over 1 ¯ iÉ ¯ n. Furthermore, each collection {e iè , . . . , e i‹} must consist of distinct basis vectors in order that T(e iè , . . . , e i‹) ≠ 0. But the antisymmetry of T tells us that for any ß ∞ Sr, we must have T(eßiè , . . . , e ßi‹) = (sgn ß)T(e iè , . . . , e i‹) where we may choose iè < ~ ~ ~ < i r. Thus, since the number of ways of choos - ing r distinct basis vectors {e iè , . . . , e i‹} out of the basis {eè, . . . , eñ} is ( rˆ), it follows that dim „ r(V) = ( rˆ) = n!/[r!(n - r)!] . We will prove this result again when we construct a specific basis for „ r(V) (see Theorem 11.8 below). In order to define linear transformations on Tr that preserve symmetry (or antisymmetry), we define t he symmetrizing mapping S: Tr ‘Tr and alter - nation mapping A: Tr ‘Tr by (ST)(vè, . . . , v r) = (1/r!)Í ß´S‹ T(v ß1 , . . . , v ßr) and (AT)(vè, . . . , v r) = (1/r!)Í ß´S‹ (sgn ß)T(v ß1 , . . . , v ßr) where T ∞ Tr (V) and vè, . . . , v r ∞ V. That these are in fact linear transforma - tions on Tr follows from the observation that the mapping T ß defined by Tß(vè, . . . , v r) = T(v ß1 , . . . , v ßr) is linear, and any linear combination of such mappi ngs is again a linear trans - 11.2 SPECIAL TYPES OF TEN SORS 557 formation . Given any ß ∞ Sr, it will be convenient for us to define the mapping ßÄ: Vr ‘ V r by ßÄ(vè, . . . , v r) = (v ß1 , . . . , v ßr) . This mapping permutes the order of the vectors in its argument, not the labels (i.e. the indices), and hence its argument must always be (v 1, v2, . . . , v r) or (w1, w2, . . . , w r) and so forth. Then for any T ∞ Tr (V) we define ßT ∞ Tr (V) by ßT = T ı ßÄ which is the mapping T ß defined above. It should be clear that ß(Tè + Tì) = ßTè + ßTì. Note also that if we write ßÄ(vè, . . . , v r) = (v ß1 , . . . , v ßr) = (wè, . . . , w r) then wá = v ßi and therefore for any other † ∞ Sr we have †Ä ı ßÄ(v 1, . . . , v r) = †Ä(w 1, . . . , w r) = (w †1, . . . , w †r) = (v ߆1, . . . , v ߆r) = `ßı† (v1, . . . , v r) . This shows that †Ä ı ßÄ = `ßı† and hence ß(†T) = ß(T ı †Ä) = T ı (†Ä ı ßÄ) = T ı (`ßı†) = (ß ı †)T . Note also that in this notation, the alternation mapping is defined as


AT = (1/r!)Í ß´S‹ (sgn ß)(ßT) . Theorem 11.3 The linear mappings A and S have the following properties: (a) T ∞ „r(V) if and only if AT = T, and T ∞ ∑r(V) if and only if ST = T. (b) A(Tr (V)) = „ r(V) and S(Tr (V)) = ∑r(V). (c) A2 = A and S2 = S, i.e., A and S are projections. Proof Since the mapping A is more useful, we will prove the theorem only for this case, and leave the analogous results for S to the reader (see Exercise 11.2.1). Furthermore, all three statements of the theorem are interrelated, so MULTILINEAR MAPPINGS AND TENSORS 558 we prove them togeth er. First suppose that T ∞ „r(V). From the definition of antisymmetric tensor we have T(v ß1 , . . . , v ßr) = (sgn ß)T(vè, . . . , v r), and thus using the fact that the order of S r is r!, we see that AT(v1,!…,!vr)=(1/r!)!"#Sr(sgn")T(v"1,!…,v"r)=(1/r!)!"#SrT(v1,!…!,!vr)=T(v1,!…!,!vr)!!. This shows that T ∞ „r(V) impl ies AT = T. Next, let T be any element of Tr (V). We may fix any particular element œ ∞ Sr and then apply AT to the vectors v œ1 , . . . , v œr to obtain AT(v!1,!…,!v!r)=AT!(v1,!…!,!vr)=(1/r!)"#$Sr(sgn#)T!(v#1,!…!,!v#r)=(1/r!)"#$Sr(sgn#)T(v#!1,!…!,!v#!r)!!. Now note that sgn ß = (sgn ß)(sgn œ)(sgn œ) = (sgn ßœ)(sgn œ), and that S r = {ƒ = ßœ: ß ∞ Sr} (this is essentially what was done in Theorem 4.3). We now see that the right hand side of the above equation is just (1/r!)(sgn!)"#$Sr(sgn#!)T(v#!1,!…!,!v#!r)=(1/r!)(sgn!)"%$Sr(sgn%)T(v%1,!…!,!v%r)=(sgn!)AT(v1,!…!,!vr) which shows (by definition) that AT is antisymmetric. In other words, this shows that AT ∞ „r(V) for any T ∞ Tr (V), or A(Tr (V)) ™ „ r(V). Since the result of the earlier paragraph showed that T = AT ∞ A(Tr (V)) for every T ∞ „r(V), we see that „ r(V) ™ A(Tr (V)), and therefore A(Tr (V)) = „ r(V). This also shows that if AT = T, then T is ne cessarily an element of „r(V). It then follows that for any T ∞ Tr (V) we have AT ∞ „r(V), and hence A2T = A(AT) = AT so that A2 = A. ˙ Suppose A iè ~ ~ ~ i‹ and T iè ~ ~ ~ i‹ (where r ¯ n = dim V and 1 ¯ i k ¯ n) are both antisymmetric tensors, and consid er their contraction A iè ~ ~ ~ i‹ Tiè ~ ~ ~ i‹ . For any particular set of indices i 1, . . . , i r there will be r! different ordered sets (i 1, . . . , i r). But by antisymmetry, the values of A iè ~ ~ ~ i‹ corresponding to each ordered set will differ only b y a sign, and similarly for T iè ~ ~ ~ i‹ . This 11.2 SPECIAL TYPES OF TEN SORS 559 means that the product of A iè ~ ~ ~ i‹ times T iè ~ ~ ~ i‹ summed over the r! ordered sets (i 1, . . . , i r) is the same as r! times a single product which we choose to be the indices i 1, . . . , i r taken in inc reasing order. In other words, we have Aiè ~ ~ ~ i‹ Tiè ~ ~ ~ i‹ = r! A \iè ~ ~ ~ i‹ \ Tiè ~ ~ ~ i‹ where \i1 ~ ~ ~ i r\ denotes the fact that we are summing over increasing sets of indices only. For example, if we have antisymmetric tensors A ijk and T ijk in ®3, then AijkTijk = 3!A \ijk\ Tijk = 6A 123T123 (where, in this case of course, A ijk and T ijk can only differ by a scalar). There is a simple but extremely useful special type of antisym metric tensor that we now wish to define. Before doing so ho wever, it is first convenient to introduce another useful notational device. Note that if T ∞ Tr and we replace vè, . . . , v r in the definitions of S and A by basis vectors eá, then we obtain an expression in terms of components as ST1 ~ ~ ~ r = (1/r!) Íß´S‹ Tß1 ~ ~ ~ ßr and


AT1 ~ ~ ~ r = (1/r!)Í ß´S‹ (sgn ß)T ß1 ~ ~ ~ ßr . We will write T (1 ~ ~ ~ r) = ST1 ~ ~ ~ r and T [1 ~ ~ ~ r] = AT1 ~ ~ ~ r . For example, we have T(ij) = (1/2!)(T ij + T ji) and T[ij] = (1/2!)(T ij - Tji) . A similar definiti on applies to any mixed tensor such as Tk(pq)[ij]=(1/2!){Tk(pq)ij!Tk(pq)ji}=(1/4){Tkpqij+Tkqpij!Tkpqji!Tkqpji}!!. Note that if T ∞ ∑r(V), then T (iè ~ ~ ~ i‹) = T iè ~ ~ ~ i‹ , while if T ∞ „r(V), then T[iè ~ ~ ~ i‹] = T iè ~ ~ ~ i‹ . Now consider the vector space ® 3 with the standard orthonormal basis {eè, eì, e 3}. We define the antisymmetric tensor ´ ∞ „3(®3) by the require - ment that !123=!(e1,!e2,!e3)=+1!!. MULTILINEAR MAPPINGS AND TENSORS 560 Since dim „ 3(®3) = 1, this defines all the components of ´ by antisymmetry: ´213 = -´231 = ´ 321 = -1 etc. If {eõá = eéa já} is any other orthonormal basis for ® 3 related to the first basis by an (orthogonal) tran sition matrix A = (a ji) with determinant equal to +1, then it is easy to see that ´(eõè, eõì, eõ 3) = det A = +1 also. This is because ´(eá, eé, eÉ) = sgn ß where ß is the permutation that takes (1, 2, 3) to (i, j, k). Since ´ ∞ „3(®3), we see that ´ [ijk] = ´ ijk. The tensor ´ is frequently called the Levi -Civita tensor . However, we stress that in a non- orthonormal coordinate system, it will not generally be true that ´ 123 = +1. While we have defined the ´ ijk as the components of a tensor, it is just as common to see the Levi -Civita (or permutation ) symbol ´ijk defined simply as an antisymmetric symbol with ´ 123 = +1. In fact, from now on we shal l use it in this way as a convenient notation for the sign of a permutation. For nota- tional consistency, we also define the permutation symbol ´ ijk to have the same values as ´ ijk. A simple calculation shows that ´ ijk ´ijk = 3! = 6. It should be clear th at this definition can easily be extended to an arbitrary number of dimensions. In other words, we define !i1!is=+1if (i1,!…!,!is) is an even permutation of (1,2,!…!,!n)"1if (i1,!…!,!is) is an odd permutation of (1,2,!…!,!n)!0otherwise#$%&%!!. This is just another way of writing sgn ß where ß ∞ Sn. Therefore, using this symbol, we have the convenient notation fo r the determinant of a matrix A = (aij) ∞ Mn(F) as det A = ´ iè ~ ~ ~ iñ aiè1 ~ ~ ~ a iñn . We now wish to prove a very useful result. To keep our notation from getting too cluttered, it will be convenient for us write !pqrijk=!pi!qj!rk . Now note that !pqr=6"pqr[123] . To see that this true, simply observe that both sides are antisymmetric in (p, q, r), and are equal to 1 if (p, q, r) = (1, 2, 3). (This also gives us a formula for ´ pqr as a 3 x 3 determi nant with entries that are a ll Kronecker delta ’s. See Exercise 11.2.4) Using ´ 123 = 1 we may write this as ´123 ´pqr = 6 !pqr[123] . But now the antisymmetry in (1, 2, 3) yields the general result !ijk!pqr=6"pqr[ijk] (1) 11.2 SPECIAL TYPES OF TEN SORS 561 which is what we wanted to show. It i s also possible to prove this in another manner that serves to illustrate several useful techniques in tensor algebra. Example 11.3 Suppose we have an arbitrary tensor A ∞ „3(®3), and hence A[ijk] = A ijk. As noted above, the fact that dim „ 3(®3) = 1 mea ns that we must have A ijk = ¬´ ijk for some scalar ¬ ∞ ®. Then Aijk!ijk!pqr="!ijk!ijk!pqr=6"!pqr=6"!ijk#pi#qj#rk!!. Because ´ ijk = ´ [ijk], we can antisymmetrize over the indices i, j, k on the right hand side of the above equation to obtain 6!"ijk#pqr[ijk] (write out the 6 terms if you do not believe it, and see Exercise 11.2.7). This gives us Aijk!ijk!pqr=6"!ijk#pqr[ijk]=6Aijk#pqr[ijk]!!. Since the antisymmetric tensor A ijk is contracted with another antisymmetric tensor on both sides of this equation, the discussion following the proof of Theorem 11.3 shows that we may write A|ijk|!ijk!pqr=6A|ijk|"pqr[ijk] or A123!123!pqr=6A123"pqr[123]!!. Cancelling A 123 then yields !123!pqr=6"pqr[123] as we had above, and hence (1) again follows from this. In the particular case that p = k, w e leave it to the reader to show that !ijk!kqr=6"kqr[ijk]="qrij#"qrji="qi"rj#"qj"ri which is very useful in manipulating vectors in ® 3. As an example, consider the vectors Aë, Bë, Cë ∞ ®3 equipped with a Cartesian coordinate system. Abusing our notation for simplicity (a lternatively, we will see formally in Example 11.12 that A i = Aá for such an Aë), we have (Bë ª Cë) i = ´áéÉB j Ck and hence Aë  (Bë ª Cë) = A i´áéÉB jCk = +´éÉáB jCkAi = Bë  (Cë ª Aë) . MULTILINEAR MAPPINGS AND TENSORS 562 Other examples are given in the exercises. ∆ Exercises 1. Prove Theorem 11.3 for the symmetrizing mapping S. 2. Using the Levi-Civita symbol, prove the following vector identities in ® 3 equipped with a Cartesian coordinate system (where the vectors are actu- ally vector fields where necessary, f is differentiab le, and # i = $ i = $/$x i): (a) Aë ª (Bë ª C ë) = (Aë  C ë)Bë - (Aë  Bë)Cë (b) (Aë ª Bë)  (Cë ª Dë) = (Aë  Cë)(Bë  Dë) - (Aë  Dë)(Bë  Cë) (c) # ª #f = 0 (d) #  (# ª Aë) = 0 (e) # ª (# ª Aë) = #(#  Aë) - #2 Aë (f ) #  (Aë ª Bë) = Bë  (# ª Aë) - Aë  (# ª Bë) (g) # ª (Aë ª Bë) = Aë(#  Bë) - Bë(#  Aë) + (Bë  #)Aë - (Aë  #)Bë 3. Using the divergence theorem (È V #  Aë d 3 x = È S Aë  nï da), prove that !"!Ad3xV#=!n"!AdaS#!!. [Hint: Let Cë be a constant vector and show that !Ci!"!Ad3xV#=(!n"!A)i!CdaS#=!Ci!n"!AdaS#!!.] 4. (a) Find the expression for ´ pqr as a 3 x 3 determinant with all Kronecker delta ’s as entries. (b) Write ´ ijk´pqr as a 3 x 3 determinant with all Kronecker delta ’s as entries. 5. Suppose V = ® 3 and let A ij be antisymmetric and S ij be sy mmetric. Show that A ijSij = 0 in two ways: (a) Write out all terms in the sum and show that they cancel in pairs. (b) Justify each of the following equalities: AijSij = A ijSji = -AjiSji = -AijSij = 0 . 6. Show that a second -rank tensor T ij can be written in the form T ij = T (ij) + T[ij], but that a third -rank tensor can not. (The complete solution for ten- 11.2 SPECIAL TYPES OF TEN SORS 563 sors of rank higher than two requires a detailed study of representations of the symmetric group and Young diagrams.) 7. Let A iè ~ ~ ~ i‹ be antisymmetric, and suppose T iè ~ ~ ~ i‹ ~ ~ ~ ~ ~ ~ is an arbitrary tensor. Show that A iè ~ ~ ~ i‹ Tiè ~ ~ ~ i‹ ~ ~ ~ ~ ~ ~ = A iè ~ ~ ~ i‹ T[iè ~ ~ ~ i‹] ~ ~ ~ ~ ~ ~ . 8. (a) Let A = (a ij) be a 3 x 3 matrix. Show ´ijk aip ajq akr = (d et A)´ pqr . (b) Let A be a linear transformation, and let y (i) = Ax (i). Show det[y (1), . . . , y (n)] = (det A) det[x (1), . . . , x (n)] . 9. Show that under orthogonal transformations A = (a ié) in ® 3, the vector cross product xï = yï ª zï transform s as xõ i = ´ ijk yõj zõk = (det A)a iéxj. Discuss the difference between the cases det A = +1 and det A = -1. 11.3 THE EXTERIOR PRODUCT We have seen that the tensor product of two elements of „ r(V) is not gen - erally another element of „ r+r(V). However, using the mapping A we can define another product on „ r(V) that turns out to be of great use. We adopt the convention of denoting elements of „ r(V) by Greek letters such as å, ∫ etc., which should not be confused with elements of the permutation group S r. If å ∞ „r(V) and ∫ ∞ „s(V), we define their exterior product (or wedge product ) å°∫ to be the mapping from „ r(V) ª „ s(V) ‘ „ r+s(V) given by !"#=(r+s)!r!s!A(!$#)!!. In other words, the wedge product is just an antisymmetrized tensor product. The reader may notice that the numerical factor is just the binomial coeffi - cient ( Âr± Í) = ( Âs± Í). It is also worth remarking that many authors leave off this coeffi cient entirely. While there is no fundamental reason for following either convention , our definition has the advantage of simplifying expressions involving volume elements as we shall se e later. MULTILINEAR MAPPINGS AND TENSORS 564 A very useful formula for computing exterior products for small values of r and s is given in the next theorem. By way of terminology, a permutation ß ∞ Sr+s such that ß1 < ~ ~ ~ < ßr and ß(r + 1) < ~ ~ ~ < ß(r + s) is called an (r, s)-shuffle . The proof of the following theorem should help to clarify this defini tion. Theorem 11.4 Suppose å ∞ „r(V) and ∫ ∞ „s(V). Then for any collection of r + s vectors vá ∞ V (with r + s ¯ dim V), we have å°∫(vè, . . . , v r+s) = Í* (sgn ß) å(v ß1 , . . . , v ßr)∫(vß(r+1) , . . . , v ß(r+s) ) where Í* denotes the sum over all permutations ß ∞ Sr+s such that ß1 < ~ ~ ~ < ßr and ß(r + 1) < ~ ~ ~ < ß(r + s) (i.e., over all (r, s) -shuffles ). Proof The proof is simply a careful examination of the terms in the definition of å° ∫. By definition, we have !"#(v1,!!…!,!vr+s)=[(r+s)!/r!s!]A(!$#)(v1,!…!,!vr+s)=[1/r!s!]%&(sgn&)!(v&1,!…!,!v&r)#(v&(r+1),!…!,!v&(r+s)) (*) where the sum is over all ß ∞ Sr+s. Now note that there are only ( Âr± Í) distinct collections {ß1, . . . , ßr}, and hence there are also only ( Âs± Í) = ( Âr± Í) distinct collections {ß(r + 1), . . . , ß(r + s)}. Let us call the set {v ß1 , . . . , vßr} the “å- variables ,” and the set {v ß(r+1) , . . . , v ß(r+s) } the “∫-variables .” For any of the ( Âr± Í) distinct collections of å - and ∫-variables , there will be r! wa ys of order - ing the å -variables within themselves, and s! ways of ordering the ∫-variables within themselves. Therefore, there will be r!s! possible arrangements of the å- and ∫-variables within themselves for each of the ( Âr± Í) distinct col lections. Let ß ∞ Sr+s be a permutation that yields one of these distinct collec tions, and assume it is the one with the property that ß1 < ~ ~ ~ < ßr and ß(r + 1) < ~ ~ ~ < ß(r + s). The proof will be finished if we can show that all the rest of the r!s! members of this collection are the same. Let T denote the term in (*) corresponding to our chosen ß. Then T is given by T = (sgn ß)å(v ß1 , . . . , v ßr)∫(vß(r+1) , . . . , v ß(r+s) ) . This means that every other term t in the distinct collection containing T w ill be of the form t = (sgn œ)å(v œ1 , . . . , v œr)∫(vœ(r+1) , . . . , v œ(r+s) ) 11.3 THE EXTERIOR PRODUCT 565 where the permutation œ ∞ Sr+s is such that the set {œ1, . . . , œr} is the same as the set {ß1, . . . , ßr} (although possibly in a different order), and similarly, the set {œ(r + 1), . . . , œ(r + s)} is the same as the set {ß(r + 1), . . . , ß(r + s)}. Thus the å - and ∫-variables are permuted within themselves. But we may then write œ = ƒß where ƒ ∞ Sr+s is again such that the two sets {ß1, . . . , ßr} and {ß(r + 1), . . . , ß(r + s)} are permuted within themselves. Because none of the transpositions that define the permutation ƒ interchange å - and ∫-variables , we may use the antisymmetry of å and ∫ separately to obtain t=(sgn!")#(v!"1,!…!,!v!"r)$(v!"(r+1),!…!,!v!"(r+s))=(sgn!")(sgn!)#(v"1,!…!,!v"r)$(v"(r+1),!…!,!v"(r+s))=(sgn")#(v"1,!…!,!v"r)$(v"(r+1),!…!,!v"(r+s))=T!!. (It was in bringing out only a single factor of sgn ƒ that we used the fact that there is no mixing of å - and ∫-variables .) In other words, the original sum over all (r + s)! possible permutations ß ∞ Sr+s has been reduced to a sum over ( Âr± Í) = (r + s)!/r!s! distinct terms, e ach one of which is repeated r!s! times. We are thus left with å°∫(vè, . . . , v r+s) = Í* (sgn ß)å(v ß1 , . . . , v ßr)∫(vß(r+1) , . . . , v ß(r+s) ) where the sum is over the (r + s)!/r!s! distinct collections {v ß1 , . . . , v ßr} and {vß(r+1) , . . . , v ß(r+s)} subject to the requirements ß1 < ~ ~ ~ < ßr and ß(r + 1) < ~ ~ ~ < ß(r + s). ˙ Let us introduce some convenient notation for handling multiple indices. Instead of writing the ordered set (iè, . . . , i r), we simply write I where the exact range w ill be clear from the context. Furthermore, we write I ö to denote the increasing sequence (iè < ~ ~ ~ < i r). Similarly, we shall also write v I instead of (v iè , . . . , v i‹). To take full advantage of this notation, we first define the generalized permuta tion symbol ´ by !i1!irj1!jr=+1if (j1,!…!,!jr) is an even permutation of (i1,!…!,!ir)"1if (j1,!…!,!jr) is an odd permutation of (i1,!…!ir)!0otherwise#$%&%!!. For example, ´ 2£3 ∞5™ = +1, ´ 3¢4£1¡ = -1, ´ 2¡3¢1™ = 0 etc. In particular, if A = (a ji) is an n x n matrix, then detA=!1!ni1!ina1i1!!anin=!i1!in1!nai11!!ainn MULTILINEAR MAPPINGS AND TENSORS 566 because !1!nj1!jn=!j1!jn=!j1!jn!!. Using this notation and Theorem 11.4, we may write the wedge product of å and ∫ as !"#(vi1,!…!,!vir+s)=$i1!ir!ir+sj1!jrk1!ks"J,"K%!(vj1,!…!,!vjr)#(vk1,!…!,!vks)! or most simply in the following form, which we state as a corollary to Theorem 11.4 for easy reference. Corollary Suppose å ∞ „r(V) and ∫ ∞ „s(V). Then for any collection of r + s vectors vá ∞ V (with r + s ¯ dim V) we have !"#(vI)=$IJK!(vJ)#(vK)!J,!K%!!. Example 11.4 Suppose dim V = 5 and {eè, . . . , e 5} is a basis for V. If å ∞ „2(V) and ∫ ∞ „1(V), then !"#(e5,!e2,!e3)=$j1<j2,k%523j1j2k!(ej1,!ej2)#(ek)=%523235!(e2,!e3)#(e5)+%523253!(e2,!e5)#(e3)+%523352!(e3,!e5)#(e2)=!(e2,!e3)#(e5)&!(e2,!e5)#(e3)+!(e3,!e5)#(e2)!!.!!! Our next theorem is a useful result in many computations. It is simply a contraction of indices i n the permutation symbols. Theorem 11.5 Let I = (iè, . . . , i q), J = (jè, . . . , j r+s), K = (kè, . . . , k r) and L = (lè, . . . , l s). Then !1!q+r+sIJ!JKL=!1!q+r+sIKL"J" where I, K and L are fixed quantities, and J is summed over all increasing subsets jè < ~ ~ ~ < j r+s of {1, . . . , q + r + s}. Proof The only nonvanishing terms on the left hand side can occur when J is a permutation of KL (or else ´ J ö ü = 0), and of these possible permutations, we only have one in the sum, and that is for th e increasing set J ö. If J is an even permutation of KL, then ´ J ö ü = +1, and ´ 1ä` ï` ` q+r+s = ´ 1ä ` ö `ü ` q+r+s since an even 11.3 THE EXTERIOR PRODUCT 567 number of permutations is required to go from J to KL. If J is an odd permu - tation of KL, then ´ J ö ü = -1, and ´ 1ä` ï` ` q+r+s = -´1ä ` ö `ü ` q+r+s since an odd number of permutations is required to go from J to KL. The concl usion then follows immediately. ˙ Note that we could have let J = (j 1, . . . , jr) and left out L entirely in Theorem 11.5. The reason we included L is shown in the next example. Example 11.5 Let us use Theorem 11.5 to give a simple proof of the asso- ciativity of the wedge product. In other words, we want to show that å°(∫°©) = (å° ∫)°© for any å ∞ „q(V), ∫ ∞ „r(V) and © ∞ „s(V). To see this, let I = (iè, . . . , i q), J = (jè, . . . , j r+s), K = (kè, . . . , k r) and L = (lè, . . . , l s). Then we h ave !"(#"$)(v1,!…!,!vq+r+s)=%!I,!J&1"q+r+sIJ!(vI)(#"$)(vJ)=%!I,!J&1"q+r+sIJ!(vI)%!K,!L&JKL#(vK)$(vL)=%!I,!K,!L&1"q+r+sIKL!(vI)#(vK)$(vL)!!. It is easy to see that had we started with (å° ∫)°©, we would have arrived at the same sum. As was the case with the tensor product, we simply write å° ∫°© from now on. Note also that a similar calculation can be done for the wedge product of any number of terms. ∆ We now wish to prove the basic algebraic properties of the wedge product. This will be facilitated by a preliminary result on the alternation mapping. Theorem 11.6 If S ∞ Tr and T ∞ Ts , then


A((AS) · T ) = A(S · T) = A(S · ( AT)) . Proof Using the bilinearity of the tensor product and the definition of AS we may write (AS) · T = (1/r!)Í ß´S‹ (sgn ß)[(ßS) · T] . For each ß ∞ Sr, let G ™ S r+s be the set of permutations ƒ defined by (ƒ1, . . . , ƒ(r + s)) = (ß1, . . . , ßr, r + 1, . . . , r + s) . MULTILINEAR MAPPINGS AND TENSORS 568 In other words, G consists of all permutations ƒ ∞ Sr+s that have the same effect on 1, . . . , r as ß ∞ Sr, but leave the remaining terms r + 1, . . . , r + s unchanged. This means that sgn ƒ = sgn ß, and ƒ(S · T) = (ßS) · T (see Exercise 11.3.1). We then have (AS) · T = (1/r!)Í ƒ´G (sgn ƒ)ƒ(S · T) and therefore A((AS)!T)=[1/(r+s)!]"#$Sr+s(sgn#)#((1/r!)"%$G(sgn%)%(S!T))=[1/(r+s)!](1/r!)"%$G"#$Sr+s(sgn#)(sgn%)#%(S!T)!!. But for each ƒ ∞ G, we note that S r+s = {œ = †ƒ: † ∞ Sr+s}, and hence [1/(r+s)!]!"#Sr+s(sgn")(sgn$)"$(S%T)=[1/(r+s)!]!"#Sr+s(sgn"$)"$(S%T)=[1/(r+s)!]!&#Sr+s(sgn&)&(S%T)=A(S%T)!!. Since this is independent of the particular ƒ ∞ G and there are r! elements in G, we then have A((AS)!T)=(1/r!)"#$GA(S!T)=A(S!T)(1/r!)"#$G1=A(S!T)!!. The proof that A(S · T) = A(S · ( AT)) is similar, and we leave it to the reader (see Exercise 11.3.2). ˙ Note that in defining the wedge product å° ∫, there is really nothing that requires us to have å ∞ „r(V) and ∫ ∞ „s(V). We could just as well be more general and let å ∞ Tr (V) and ∫ ∞ Ts (V). However, if this is the case, then the formula given in Theorem 11.4 most certainly is not valid. However, we do have the following corollary to Theorem 11.6. Corollary For any S ∞ Tr and T ∞ Ts we have AS°T = S°T = S° AT. Proof This follows directly from Theorem 11.6 and the wedge product defini tion S°T = [(r + s)!/r!s!] A(S · T). ˙ We are now in a position to prove some of the most important properties of the wedge product. Note that this next theorem is stated in terms of the 11.3 THE EXTERIOR PRODUCT 569 more general definition of the wedge product. Theorem 11.7 Suppose å, åè, åì ∞ Tq (V), ∫, ∫è, ∫ì ∞ Tr (V), © ∞ Ts (V) and a ∞ F. Then (a) The wedge product is bilinear. That is, (åè + åì)° ∫ = åè° ∫ + åì° ∫ å°( ∫è + ∫ì) = å° ∫è + å° ∫ì (aå)° ∫ = å°(a ∫) = a(å° ∫) (b) å° ∫ = (-1)qr∫°å. (c) The wedge product is associative. That is, å°( ∫°©) = (å° ∫)°© = [(q + r + s)!/q!r!s!] A(å · ∫ · ©) . Proof (a) This follows from the definition of wedge product, the fact that · is bilinear and A is linear. This result may also be shown directly in the case that å, åè, åì ∞ „q(V) and ∫, ∫è, ∫ì ∞ „r(V) by using the corollary to Theorem 11.4 (see Exercise 11.3.3). (b) This can also be shown directly from the corollary to Theorem 11.4 (see Exercise 11.3.4). Alternatively, we may proceed as follows. First note that for ß ∞ Sr, we see that (since for a ny other † ∞ Sr we have †(ßå) = († ı ß)å, and hence †(ßå)(vè, . . . , v r) = å(v †ß1 , . . . , v †ßr)) A(!")(v1,!…!,!vr)=(1/r!)#$%Sr(sgn$)$(!")(v1,!…!,!vr)=(1/r!)#$%Sr(sgn$)"(v$!1,!…!,!v$!r)=(1/r!)#$%Sr(sgn$!)(sgn!)"(v$!1,!…!,!v$!r)=(sgn!)(1/r!)#&%Sr(sgn&)"(v&1,!…!,!v&r)=(sgn!)A!(v1,!…!,!vr)!!. Hence A(ßå) = (sgn ß) Aå. Now define ßà ∞ Sq+r by ßà(1, . . . , q + r) = (q + 1, . . . , q + r, 1, . . . , q) . Since ßà is just a product of qr transpositions, it follows that sgn ßà = ( -1)qr. We then see that !"#(v1,!…!,!vq+r)=(#"!)(v$01,!…!,!v$0(q+r))=$0(#"!)(v1,!…!,!vq+r)!!. MULTILINEAR MAPPINGS AND TENSORS 570 Therefore (ignoring the factorial multiplier which will cancel out from both sides of this equation), !"#=A(!$#)=A(%0(#$!))=(sgn%0)A(#$!)=(&1)qr#"!!!. (c) Using Theorem 11.6, we simply calculate (!"#)"$=[(q+r+s)!/(q+r)!s!]A((!"#)%$)=[(q+r+s)!/(q+r)!s!][(q+r)!/q!r!]A(A(!%#)%$)=[(q+r+s)!/(q!r!s!]A(!%#%$)!!. Similarly, we find that å°( ∫°©) yields the same result. We are therefore justi - fied (as we also saw in Example 11.5) in writing simply å° ∫°©. Furthermore, it is clear that this result can be extended to any finite number of products. ˙ Example 11.6 Sup pose å ∞ Tr and ∫ ∞ Ts. Since å° ∫ = (-1)rs∫°å, we see that if either r or s is even, then å° ∫ = ∫°å, but if both r and s are odd, then å°∫ = -∫°å. Therefore if r is odd we have å°å = 0, but if r is even, then å°å is not necessarily zero. In particular, any 1-form å always has the property that å°å = 0. ∆ Example 11.7 If å 1 , . . . , å 5 are 1-forms on ® 5, let us define ∫ = å 1°å3 + å 3°å5 and © = 2å 2°å4°å5 - å1°å2°å4 . Using the properties of the wedge product given in Theorem 11.7 we then have ∫°© = (å 1°å3 + å 3°å5)°(2å 2°å4°å5 - å1°å2°å4) = 2å 1°å3°å2°å4°å5 - å1°å3°å1°å2°å4 + 2å 3°å5°å2°å4°å5 - å3°å5°å1°å2°å4 = -2å1°å2°å3°å4°å5 - 0 + 0 - å1°å2°å3°å4°å5 = -3å1°å2°å3°å4°å5 . ∆ Example 11.8 Suppose å 1 , . . . , å r ∞ „1(V) and vè, . . . , v r ∞ V. Using Theorem 11.5, it is easy to generalize the corollary to Theorem 11.4 to obtain (see Exercise 11.3.5) 11.3 THE EXTERIOR PRODUCT 571 !1"!"!r(v1,…,vr)=#i1!ir$1!ri1!ir!1(vi1)!!r(vir)=det(!i(vj))!!. (Note that the sum is not over any increasing indices because each å i is onl y a 1-form .) As a special case, suppose {eá} is a basis for V and {ø j} is the cor- responding dual basis. Then ø j(eá) = ∂já and hence !i1"!"!ir(ej1,…,!ejr)=#k1!kr$j1!jrk1!kr!i1(ek1)!!ir(ekr)=$j1!jri1!ir!!. In particular, if dim V = n, choosing the indices (iè, . . . , i n) = (1, . . . , n) = (jè, . . . , j n), we see that ø 1° ~ ~ ~ °øn (eè, . . . , eñ) = 1 . ∆ Exercises 1. Show that ƒ(S · T) = (ßS) · T in the proof of Theorem 11.6. 2. Finish the proof of Theorem 11.6 by showing that A(S · T) = A(S · ( AT)). 3. Using åá ∞ „q(V) and ∫á ∞ „r(V), prove Theorem 11.7(a) directly from the corollary to Theorem 11.4. 4. Use the corollary to Theorem 11.4 to prove Theorem 11.7(b). 5. Suppose åè, . . . , å r ∞ „1(V) and vè, . . . , v r ∞ V. Show that åè° ~ ~ ~ °å r(vè, . . . , v r) = det(åá(v é)) . 6. Suppose {eè, . . . , eñ} is a basis for V and {ø 1, . . . , ø n} is the corre - sponding dual basis. If å ∞ „r(V) (where r ¯ n), show that å = Í Iö å(e I)øI = Í iè< ~ ~ ~ <i‹ å(e iè, . . . , e i‹)øiè° ~ ~ ~ °øi‹ by applying both sides to (e jè, . . . , e j‹). (See also Theorem 11.8.) MULTILINEAR MAPPINGS AND TENSORS 572 7. (Interior Product ) Suppose å ∞ „r(V) and v, vì, . . . , v r ∞ V. We define the (r - 1)-form ivå by iv!=0if r=0.iv!=!(v)if r=1.iv!(v2,!…!,!vr)=!(v,!v2,!…!,!vr)if r>1. (a) Prove that iu+v = iu + iv. (b) If å ∞ „r(V) and ∫ ∞ „s(V), prove that iv: „ r+s(V) ‘ „ r+s-1(V) is an anti-derivation , i.e., iv(å°∫) = ( iv å)°∫ + (-1)rå°(iv ∫) . (c) If v = v ieá and å = Í Iö aiè ~ ~ ~ i‹ øiè° ~ ~ ~ °øi‹ where {ø i} is the basis dual to {e i}, show that ivå = Í iì< ~ ~ ~ <i‹ biì ~ ~ ~ i‹ øiì°~ ~ ~°ø i‹ where biì ~ ~ ~ i‹ = Íév j ajiì ~ ~ ~ i‹ . (d) If å = f 1° ~ ~ ~ °fr, show that iv!=("1)k"1fk(v)f1#!#fk"1#fk+1#!#frk=1r$=("1)k"1fk(v)f1#!#fk"k=1r$#!#fr where the Ä means that the term f k is to be deleted from the expression. 8. Let V = ® n have the sta ndard basis {eá}, and let the corresponding dual basis for V* be {ø i}. (a) If u, v ∞ V, show that !i"!j(u,!v)=uiviujvj and that this is ± the area of the parallelogram spanned by the projection of u and v onto the x i xj-plane . What do you think is the significance of the different signs? 11.3 THE EXTERIOR PRODUCT 573 (b) Generalize this to ø iè° ~ ~ ~ °øi‹ where r ¯ n. 9. Suppose V = Fn, and let v 1, . . . , v n ∞ V have components relative to the standard basis {eá} defined by v i = e jvji. For any 1 ¯ r < n, let s = n - r and define the r-form å by !(v1,!…!,!vr)=v11!v1r""vr1!vrr!!! and the s-form ∫ by !(v1,!…!,!vs)=v1r+1!v1n""vsr+1!vsn!!. (a) Use Theorem 4.9 to show that å° ∫ is the determinant function D on Fn. (b) Show that the sign of an (r, s) -shuffle is given by !1!r!r+1!r+si1!irj1!js=("1)i1+!+ir+r(r+1)/2 where i 1, . . . , i r and j 1, . . . , j s are listed in increa sing order. (c) If A = (a ij) ∞ Mn(F), prove the Laplace expansion formula detA=!!I("1)i1+"+ir+r(r+1)/2ai11"ai1r##air1"airraj1r+1"aj1n##ajrr+1"ajrn where I = {i 1, . . , i r} and J = {j 1, . . . , j s} are “complementary ” sets of indices, i.e., I ⁄ J = Å and I ¡ J = {1, 2, . . . , n}. 10. Let B = r!A where A: Tr ‘Tr is the alternation mapping. Define å° ∫ in terms of B. What is B(f1 · ~ ~ ~ · f r) where f i ∞ V*? 11. Let I = ( i1, . . . , iq), J = (j 1, . . . , jp), and K = (k 1, . . . , kq). Prove the following generalization of E xample 11.3: !1!p+qJI!JK1!p+q"J"=!KI=n!#k1[i1!#kqiq] MULTILINEAR MAPPINGS AND TENSORS 574 11.4 TENSOR ALGEBRAS We define the direct sum of all tensor spaces Tr (V) to be the ( infinite - dimensional ) space T0 (V) • T1(V) • ~ ~ ~ • Tr (V) • ~ ~ ~ , and T (V) to be all elements of this space with finitel y many nonzero compo nents . This means that every element T ∞ T(V) has a unique expression of the form (ignoring zero summands) T = T (1)iè + ~ ~ ~ + T (r)i‹ where each T (k)iÉ ∞ TiÉ (V) and iè < ~ ~ ~ < i r. The tensors T (k)iÉ are called the graded components of T. In the special case that T ∞ Tr (V) for some r, then T is said to be of order r. We define addition in T (V) componentwise, and we also define multiplication in T (V) by defining · to be distributive on all of T (V). We have therefore made T (V) into an associative algebra over F which is called the tensor algebra . We have seen that „ r(V) is a subspace of Tr (V) since „ r(V) is just the image of Tr (V) under A. Recall also that „ 0(V) = T0 (V) is defined to be the scalar fi eld F. As might therefore be expected, we define „(V) to be the direct sum „(V) = „ 0(V) • „ 1(V) • ~ ~ ~ • „ r(V) • ~ ~ ~ ™ T (V) . Note that „ r(V) = 0 if r > dim V. It is important to realize that if å ∞ „r(V) ™ Tr (V) and ∫ ∞ „s(V) ™ Ts (V), then even though å · ∫ ∞ Tr+s (V), it is not generally true that å · ∫ ∞ „r+s(V). Therefore „(V) is not a subalgebra of T (V). However, the wedge product is a mapping from „ r(V) ª „ s(V) ‘ „ r+s(V), and hence if we extend this product in the obvious manner to a bilinear mapping „(V) ª „(V) ‘ „(V), then „(V) becomes an algebra over F = „ 0(V). In other words, if å = åè + ~ ~ ~ + å r with each åá ∞ „rá(V), and ∫ = ∫è + ~ ~ ~ + ∫s with each ∫á ∞ „sá(V), then we define !"#=!i"#jj=1s$i=1r$!!. This algebra is called the Grassmann (or exterior ) algebra . The astute reader may be won dering exactly how we add together the ele- ments åè ∞ „rè(V) and åì ∞ „rì(V) (with rè ≠ rì) when none of the opera - tions (åè + åì)(vè, . . . , v rè), (åè + åì)(vè, . . . , v rì) nor (åè + åì)(vè, . . . , vrè+rì) makes any sense. The answer is that for purpos es of the Grassmann algebra, we consider both åè and åì to be elements of „(V). For example, if 11.4 TENSOR ALGEBRAS 575 åè is a 1-form and åì is a 2-form , then we write åè = 0 + åè + 0 + 0 + ~ ~ ~ and åì = 0 + 0 + åì + 0 + ~ ~ ~ , and hence aèåè + aìåì (where aá ∞ F) makes sense in „(V). In this way, every element of „(V) has a degree (recall that an r- form is said to be of degree r), and we say that „(V) is a graded associative algebra . Unlike the infinite -dimensional algebra T (V), the algebr a „(V) is finite - dimensional . This should be clear from the discussion following Example 11.2 where we showed that dim „ r(V) = ( rˆ) (where n = dim V), and hence that dim „r(V) = 0 if r > n. Let us now prove this result again by constructing a specific basi s for „ r(V). Theorem 11.8 Suppose dim V = n. Then for r > n we have „ r(V) = {0}, and if 0 ¯ r ¯ n, then dim „ r(V) = ( rˆ). Therefore dim „(V) = 2 n. Moreover, if {ø1, . . . , ø n} is a basis for V* = „ 1(V), then a basis for „ r(V) is given by the set {ø iè° ~ ~ ~ °øi‹: 1 ¯ iè < ~ ~ ~ < i r ¯ n} . Proof Suppose å ∞ „r(V) where r > dim V = n. By multilinearity, å is deter mined by its values on a basis {eè, . . . , eñ} for V (see Example 11.2). But then we must have å(e iè , . . . , e i‹) = 0 because at least two of the e iÉ are necessar ily the same and å is antisymmetric. This means that å(vè, . . . , v r) = 0 for all vá ∞ V, and hence å = 0. Thus „ r(V) = {0} if r > n. Now suppose that {ø 1, . . . , ø n} is the basis for V* dual to {eá}. From Theorem 11.2, we know that {ø iè · ~ ~ ~ · ø i‹: 1 ¯ iè , . . . , i r ¯ n} forms a basis for Tr (V), and since the alternation mapping A maps Tr (V) onto „ r(V) (Theorem 11.3(b)), it follows that the image of the basis {ø iè · ~ ~ ~ · ø i‹} for Tr (V) must span „ r(V). If å ∞ „r(V), then å ∞ Tr (V), and hence å = å iè ~ ~ ~ i‹ øiè · ~ ~ ~ · ø i‹ where the sum is over all 1 ¯ iè, . . . , i r ¯ n and å iè ~ ~ ~ i‹ = å(e iè , . . . , e i‹). Using Theorems 11.3(a) and 11.7(c) we have !=A!=!i1!irA("i1#!#"ir)=!i1!ir(1/r!)"i1$!$"ir where the sum is still over all 1 ¯ iè, . . . , i r ¯ n. However, by the antisymme - try of the wedge product, the collection {iè, . . . , i r} must all be different, and hence the sum can only be over the ( rˆ) distinct such combinations. For each MULTILINEAR MAPPINGS AND TENSORS 576 such combination there will be r! permu tations ß ∞ Sr of the basis vectors. If we write each of these permutations in increasing order iè < ~ ~ ~ < i r, then the wedge product changes by a factor sgn ß, as does å iè ~ ~ ~ i‹ = å(e iè , . . . , e i‹). Therefore the signs cancel and we are left with å = å \iè ~ ~ ~ i‹ \ øiè° ~ ~ ~ °øi‹ where, as mentioned previously, we use the notation å \ iè ~ ~ ~ i‹ \ to mean that the sum is over increasing sets iè < ~ ~ ~ < i r. Thus we have shown that the ( rˆ) ele- ments ø iè° ~ ~ ~ °ø i‹ with 1 ¯ iè < ~ ~ ~ < i r ¯ n span „ r(V). We must still show that they are linearly independent. Suppose å \iè ~ ~ ~ i‹ \ øiè° ~ ~ ~ °øi‹ = 0. Then for any set {e jè , . . . , e j‹} with 1 ¯ jè < ~ ~ ~ < j r ¯ n we have (using Example 11.8) 0=!|i1!ir|"i1#!#"ir(ej1,…,!ejr)=!|i1!ir|$j1!jri1!ir=!j1!jr since the only non vanishing term occurs when {iè, . . . , i r} is a permutation of {jè, . . . , j r} and both are increasing sets. This proves linear independence. Finally, using the binomial theorem, we now see that dim„(V)=dim„r(V)r=0n!=nr"#$%&'r=0n!=(1+1)n=2n!!.!!˙ Example 11.9 Another usefu l result is the following. Suppose dim V = n, and let {ø 1, . . . , ø n} be a basis for V*. If å 1, . . . , å n are any other 1-forms in „1(V) = V*, then we may expand each å i in terms of the ø j as å i = a iéøj. We then have !1"!"!n=a1in!anin#i1"!"#in=a1in!anin$1!ni1!in#1"!"#n=det(aij)#1"!"#n!!. Recalli ng Example 11.1, if {ø i = dx i} is a local basis for a cotangent space V* and {å i = dy i} is any other local basis, then dy i = ($y i/$xj)dxj and 11.4 TENSOR ALGEBRAS 577 det(aij)=!(y1!yn)!(x1!xn) is just the usual Jacobian of the transformation. We then have dy1!!!dyn="(y1!yn)"(x1!xn)dx1!!!dxn!!. The reader may recognize dx 1° ~ ~ ~ °dx n as the volume element on ® n, and hence differential forms are a natural way to descr ibe the change of variables in multiple integrals. ∆ Theorem 11.9 If å 1, . . . , å r ∞ „1(V), then {å 1, . . . , å r} is a linearly dependent set if and only if å 1° ~ ~ ~ °år = 0. Proof If {å 1, . . . , å r} is linearly dependent, then there exists at le ast one vector, say å 1, such that !1="j#1aj!j . But then !1"!"!r=(#j$1aj!j)"!2"!"!r=#j$1aj(!j"!2"!"!r)=0 since every term in the sum contains a repeated 1-form and hence vanishes. Conversely, suppose that å 1, . . . , å r are linearly independent . We can then extend them to a basis {å 1, . . . , å n} for V* (Theorem 2.10). If {eá} is the corresponding dual basis for V, then å 1° ~ ~ ~ °ån(eè, . . . , eñ) = 1 which implies that å 1° ~ ~ ~ °år ≠ 0. Therefore {å 1, . . . , å r} must be linearly depen - dent if å 1° ~ ~ ~ °år = 0. ˙ 11.5 THE TENSOR PRODUCT OF VECTOR SPACES We now discuss the notion of the tensor product of vector spaces. Our reason for presenting this discussion is that it provi des the basis for defining the Kronecker (or direct) product of two matrices, a concept which is very useful in the theory of group representations. It should be remarked that there are many ways of defining the tensor product of vector spaces. While we w ill follow the simplest approach, there is another (somewhat complicated) method involving quotient spaces that is also MULTILINEAR MAPPINGS AND TENSORS 578 frequently used. This other method has the advantage that it includes infinite - dimensional spaces. The reader can find a treatment of t his alternative method in, e.g., in the book by Curtis (1984). By way of nomenclature, we say that a mapping f: U ª V ‘ W of vector spaces U and V to a vector space W is bilinear if f is linear in each variable. This is exactly t he same as we defined in Section 9.4 except that now f takes its values in W rather than the field F. In addition, we will need the concept of a vector space generated by a set. In other words, suppose S = {sè, . . . , sñ} is some finite set of objects, an d F is a field. While we may have an intuitive sense of what it should mean to write formal linear combinations of the form aèsè + ~ ~ ~ + añsñ, we should realize that the + sign as used here has no meaning for an arbitrary set S. We now go through the fo rmalities involved in defining such terms, and hence make the set S into a vector space T over F. The basic idea is that we want to recast each element of S into the form of a function from S to F. This is because we already know how to add functions as well as multiply them by a scalar. With these ideas in mind, for each sá ∞ S we define a function sá: S ‘ F by sá(sé) = 1 ∂áé where 1 is the multiplicative identity of F. Since addition in F is well-defined as is the addition of functions and multiplica tion of functions by elements of F, we see that for any a, b ∞ F and sá, sé ∞ S we have (a+b)si(sj)=(a+b)!ij=a!ij+b!ij=asi(sj)+bsi(sj)=(asi+bsi)(sj) and therefore (a + b)sá = asá + bsá. Similarly, it is easy to see that a(bsá) = (ab)sá. We now define T to be the set of all functions fr om S to F. These func- tions can be written in the form aèsè + ~ ~ ~ + añsñ with aá ∞ F. It should be clear that with our definition of the terms aásá, T forms a vector space over F. In fact, it is easy to see that the functions 1sè , . . . , 1sñ are linea rly independent. Indeed, if 0 denotes the zero function, suppose aèsè + ~ ~ ~ + añsñ = 0 for some set of scalars aá. Applying this function to sá (where 1 ¯ i ¯ n) we obtain aá = 0. As a matter of course, we simply write sá rather than 1sá. The linear com binations just defined are called formal linear combina - tions of the elements of S, and T is the vector space generated by the set S. T is therefore the vector space of all such formal linear combinations, and is sometimes called the free vector space of S over F. 11.5 THE TENSOR PRODUCT O F VECTOR SPACES 579 Theorem 11.10 Let U, V and W be finite -dimensional vector spaces over F. Then there exists a finite -dimensional vector space over F denoted by T a nd a bilinear mapping t: U ª V ‘ T denoted by t(u, v) = u · v satisfy ing the fol - lowing properties: (a) For every bilinear mapping f: U ª V ‘ W, there exists a unique linear transformation f ÿ: T ‘ W such that f = f ÿ ı t. In other words, for all u ∞ U and v ∞ V we have f(u, v) = f ÿ(t(u, v)) = f ÿ(u · v) . (b) If {uè, . . . , u m} is a basis for U and {vè, . . . , vñ} is a basis for V, then {uá · vé} is a basis for T and therefore dim T = mn = (dim U)(di m V) . Proof Let {uè, . . . , u m} be a basis for U and let {vè, . . . , vñ} be a basis for V. For each pair of integers (i, j) with 1 ¯ i ¯ m and 1 ¯ j ¯ n we let táé be a letter (i.e., an element of some set). We now define T to be the vector space over F consisting of all formal linear combina tions of the elements táé. In other words, every element of T is of the form a ij táé where a ij ∞ F. Define the bilinear map t: U ª V ‘ T by ui · v j ^ t(u i, vj) = t ij and hence to all of U ª V by “bilinear extension. ” In particular, if u = x iuá ∞ U and v = y jvé ∞ V, let us define u · v to be that element of T given by u · v = t(u, v) = xiyj táé . It should be obvious that this does indeed define a bilinear map. Now suppose that f: U ª V ‘ W is any bilinear map, and remember that every element of T is a linear combination of the táé. According to Theorem 5.1, we may define a unique l inear transformation f ÿ: T ‘ W by f ÿ(táé) = f(uá, vé) . Using the bilinearity of f and the linearity of f ÿ we then have f(u,!v)=f(xiui,!yjvj)=xiyjf(ui,!vj)=xiyj!f(tij)=!f(xiyjtij)=!f(u!v)=!f(t(u,!v))!!. MULTILINEAR MAPPINGS AND TENSORS 580 This proves the existence and uniqueness of the mapping f ÿ such that f = f ÿı t as specified in (a). We have defined T to be the vector space generated by the mn elements táé = uá · vé where {uè, . . . , u m} and {vè, . . . , vñ} were particular bases for U and V respectively. We now want to show that in fact {uæá · væé} forms a basis for T where {uæá} and {væé} are arbitrary bases for U and V. For any u = xæi uæá ∞ U and v = yæ j væé ∞ V, we have (using the bilinearity of ·) u · v = xæ i yæj (uæá · væé) which shows that the mn elements uæá · væé span T. If these mn elements are linearly dependent, then dim T < mn which contradicts the fact that the mn elements táé form a basis for T. Hence {uæá · væé} is a basis for T. ˙ The space T defined in this theorem is denoted by U · V and called the tensor product of U and V. Note that T can be any mn dimensional vector space. For example, if m = n we could take T = T 2(V) with basis t ij = ø i · ø j, 1 ¯ i, j ¯ n. The map t(u i, vj) = u i · v j then defines u i · v j = ø i · ø j. Example 11.10 To show how this formalism relates to our previous treat- ment of tensors, consider the following example of the mapping f ÿ defined in Theorem 11.10. Let {eá} be a basis for a real inner product space U, and let us define the real numbers gáé = Óeá, eéÔ. If eõá = eépjá is another basis for U, then gõáé = Óeõá, eõéÔ = p rápséÓer , esÔ = p rápsé grs so that the gáé transform like the components of a covariant tensor of order 2. This means that we may define the tensor g ∞ T2(U) by g(u, v) = Óu, vÔ. This tensor is called the metric tensor on U (see Section 11.10). Now suppose that we are given a positive definite symmetric bilinear form (i.e., an inner product) g = Ó , Ô: U ª U ‘ F. Then the mapping gÿ is just the metric because g ÿ(eá · eé) = g(eá, eé) = Óeá, eéÔ = gáé . Therefore, if u = u ieá and v = v jeé are vectors in U, we see that g ÿ(u · v) = g(u, v) = Óu, vÔ = u i vj Óeá, eéÔ = gáéu i vj . If {ø i} is the basis for U* dual to {eá}, then according to our earlier formal - ism, we would write this as g ÿ = gáéø i · ø j. ∆ 11.5 THE TENSOR PRODUCT O F VECTOR SPACES 581 Some of the main applications in mathematics and physics (e.g., in the theory of group representations) of the tensor product of two vector spaces are contained in the next two results. While the ide as are simple enough, the nota- tion becomes somewhat awkward because of the double indices. Theorem 11.11 Let U and V have the respective bases {uè, . . . , u m} and {vè, . . . , vñ}, and suppose the linear operators S ∞ L(U) and T ∞ L(V) have matrix rep resentations A = (a ié) and B = (b ié) respectively. Then there exists a linear trans formation S · T: U · V ‘ U · V such that for all u ∞ U and v ∞ V we have (S · T)(u · v) = S(u) · T(v). Furthermore, the matrix C of S · T relative to the ordered basis {uè · vè, . . . , uè · vñ, uì · vè, . . . , uì · vñ, . . . , u m · vè, . . . , u m · vñ} for U · V is the mn x mn matrix given in block matrix form as C=a11Ba12B!a1mB"""am1Bam2B!ammB!"###$%&&&!!. The matrix C is called the Kronecker (or direct or tensor ) product of the matrices A and B, and will also be written as C = A · B. Proof Since S and T are linear and · is bilinear, it is easy to see that the mapping f: U ª V ‘ U · V defined by f(u, v) = S(u) · T(v) is bilinear. Therefore, according to Theorem 11.10, there exists a unique linear transfor - mation f ÿ ∞ L(U · V) such that f ÿ(u · v) = S(u) · T(v). We denote the map- ping f ÿ by S · T. Thus, (S · T)(u · v) = S(u) · T(v). To find the matrix C of S · T is straightforward enough. We have S(uá) = ué ajá and T(vá) = véb já, and hence (S · T)(uá · vé) = S(uá) · T(vé) = a ribsj(ur · v s) . Now recall that the i th column of the matrix representation of an operator is just the image of the i th basis vector under the transformation (see Theorem 5.11). In the present case, we will have to use double pairs of subscripts to label the matrix elements. Relative to the ordered basis {uè · vè, . . . , uè · vñ, uì · vè, . . . , uì · vñ, . . . , u m · vè, . . . , u m · vñ} MULTILINEAR MAPPINGS AND TENSORS 582 for U · V, we then see that, for example, the (1, 1) th column of C is the vector (S · T)(uè · vè) = a rèbsè(ur · v s) given by (a11b11, . . . , a 11bn1, a21b11, . . . , a 21bn1, . . . , a m1b11, . . . , a m1bn1) and in general, the (i, j) th column is given by (a 1ib1j, . . . , a 1ibnj, a2ib1j, . . . , a 2ibnj, . . . , a mib1j, . . . , a mibnj) . This shows that the matrix C has the desired form. ˙ Theorem 11.12 Let U and V be finite -dimensional vector spaces over F. (a) If Sè, Sì ∞ L(U) and Tè, Tì ∞ L(V), then (Sè · Tè)(Sì · Tì) = SèSì · TèTì . Moreover, if Aá and Bá are the matrix r epresentations of Sá and Tá respectively (relative to some basis for U · V), then (Aè · Bè)(Aì · Bì) = AèAì · BèBì. (b) If S ∞ L(U) and T ∞ L(V), then Tr(S · T) = (Tr S)(Tr T). (c) If S ∞ L(U) and T ∞ L(V), and if Sî and Tî exist, then (S · T)î = Sî · Tî . Conversely, if (S · T)î exists, then Sî and Tî also exist, and (S · T)î = Sî · Tî. Proof (a) For any u ∞ U and v ∞ V we have (S1!T1)(S2!T2)(u!v)=(S1!T1)(S2(u)!T2(v))=S1S2(u)!T1T2(v)=(S1S2!T1T2)(u!v)!!. As to the matrix representations, simply note that Aá · Bá is the representation of Sá · Tá, and AèAì · BèBì is the representation of SèSì · TèTì (since the representation of a product of linear operators is the product of their matrices). (b) Recall that the trace of a linear operator is defined to be the trace of any matrix representati on of the operator (see Theorem 5.19). Therefore, if A = (a ié) is the matrix of S and B = (b ié) is the matrix of T, we see from Theorem 11.11 that the diagonal blocks of A · B are the matrices a 1èB, . . . , ammB and hence the diagonal elements of A · B are a1èb1è, . . . , a 1èbnñ, . . . , ammb1è, . . . , a mmbnñ. Therefore the sum of these diagonal elements is just 11.5 THE TENSOR PRODUCT O F VECTOR SPACES 583 Tr(A!B)=a11("ibii)+!+amm("ibii)=("jajj)("ibii)=(TrA)(TrB)!!. (c) We first note that if 1 denotes the identity transformation, then (1 · 1)(u · v) = u · v and hence 1 · 1 = 1. Next note that u · v = (u + 0) · v = u · v + 0 · v, and hence 0 · v = 0. Similarly, it is clear that u · 0 = 0. This then shows that (S · 0)(u · v) = S(u) · 0 = 0 so that S · 0 = 0, and similarly 0 · T = 0. Now, if S and T are invertible, then by part (a) we see that (Sî · Tî)(S · T) = SSî · TîT = 1 · 1 = 1 and similarly for (S · T)(Sî · Tî). Therefore (S · T)î = Sî · Tî. Conversely, suppose that S · T is invertible. To prove that S and T are also invertible we use Theorem 5.9. In other wo rds, a surjective linear operator is invertible if and only if its kernel is zero. Since S · T is invertible we must have T ≠ 0, and hence there exists v ∞ V such that T(v) ≠ 0. Suppose u ∞ U is such that S(u) = 0. Then 0 = S(u) · T(v) = (S · T)(u · v) which implies that u · v = 0 (since S · T is invertible). But v ≠ 0, and hence we must have u = 0. This shows that S is invertible. Similarly, had we started with S ≠ 0, we would have found that T is invertible. ˙ Exercises 1. Give a direct proof of the matrix part of Theorem 11.12(a) using the defini tion of the Kronecker product of two matrices. 2. Suppose A ∞ L(U) and B ∞ L(V) where dim U = n and dim V = m. Show that det(A · B) = (det A) m (det B) n . MULTILINEAR MAPPINGS AND TENSORS 584 11.6 VOLUMES IN ®3 Instead of starting out with an abstract presentation of volumes, we shall first go through an intuitive elementary discussion beginning with ® 2, then going to ® 3, and finally generalizing to ® n in the next section. First consider a p arallelogram in ® 2 (with the usual norm) defined by the vectors X and Y as shown. Note that h = ˜ Y ˜ sin œ and b = ˜ Y ˜ cos œ, and also that the area of each tri- angle is given by Aè = (1/2)bh. Then the area of the rectangle is given by Aì = ( ˜X ˜ - b)h, and the area of the entire parallelogram is given by A=2A1+A2=bh+(˜X˜!!!!b)h=!˜X˜h=!˜X˜!˜Y˜sin"!!!!. (1) The rea der should recognize this as the magnitude of the elementary “vector cross product ” X ª Y of the ordered pair of vectors (X, Y) that is defined to have a direction normal to the plane spanned by X and Y, and given by the “right hand rule ” (i.e., out of the plane in this case). If we define the usual orthogonal coordinate system with the x-axis parallel to the vector X, then X = (x 1, x2) = ( ˜ X ˜, 0) and Y = (y 1, y2) = ( ˜ Y ˜ cos œ, ˜ Y ˜ sin œ) and hence we see that the determinant with columns formed from the vectors X and Y is just x1y1x2y2=˜X˜˜Y˜!cos!0˜Y˜!sin!=˜X˜!˜Y˜!sin!=A!!. (2) Notice that if we interchanged t he vectors X and Y in the diagram, then the determinant would change sign and the vector X ª Y (which by definition has a direction dependent on the ordered pair (X, Y)) would point into the page. h b ! X X Y Y A 2 A 1 A 1 11.6 VOLUMES IN ®3 585 Thus the area of a parallelogram (which is always positive by definition) defined by two vectors in ® 2 is in general given by the absolute value of the determinant (2). In terms of the usual inner product (or “dot product ”) Ó , Ô on ® 2, we have ÓX, XÔ = ˜ X ˜2 and ÓX, YÔ = ÓY, XÔ = ˜ X ˜ ˜Y ˜ cos œ, and hence A2=˜X˜2˜Y˜2sin2!=˜X˜2˜Y˜2(1"cos2!)=˜X˜2˜Y˜2"ÓX,!YÔ2!!. Therefore we see that the area is also given by the positive square root of the determinant A2=ÓX,!XÔÓX,!YÔÓY,!XÔÓY,!YÔ!!. (3) It is also worth noting that the inner product may be written in the form ÓX, YÔ = x 1y1 + x 2y2, and thus in terms of matrices we may write ÓX,!XÔÓX,!YÔÓY,!XÔÓY,!YÔ!"#$%&=x1x2y1y2!"##$%&&x1y1x2y2!"##$%&&!!. Hence taking the determinant of this e quation (using Theorems 4.8 and 4.1), we find (at least in ® 2) that the determinant (3) also implies that the area is given by the absolute value of the determinant in equation (2). It is now easy to extend this discussion to a parallelogram in ® 3. Indeed , if X = (x 1, x2, x3) and Y = (y 1, y2, y3) are vectors in ® 3, then equation (1) is unchanged because any two vectors in ® 3 define the plane ® 2 spanned by the two vectors. Equation (3) also remains unchanged since its derivation did not depend on the specif ic coordinates of X and Y in ® 2. However, the left hand part of equation (2) does not apply (although we will see below that the three - dimensional version determines a volume in ® 3). As a final remark on parallelograms, note that if X and Y are linearly dependent, then aX + bY = 0 so that Y = -(a/b)X, and hence X and Y are co- linear . Ther efore œ equals 0 or π so that all equations for the area in terms of sin œ are equal to zero. Since X and Y are dependent, this also means that the determinant in equation (2) equals zero, and everything is consistent. MULTILINEAR MAPPINGS AND TENSORS 586 We now take a look at volumes in ®3. Consider three linearly independent vectors X = (x 1, x2, x3), Y = (y 1, y2, y3) and Z = (z 1, z2, z3), and consider the parallelepiped with edges defined by these three vectors (in the given order (X, Y, Z)). We claim that the volume of this parallelepiped is given by both the positive square root of the determinant ÓX,!XÔÓX,!YÔÓX,!ZÔÓY,!XÔÓY,!YÔÓY,!ZÔÓZ,!XÔÓZ,!YÔÓZ,!ZÔ (4) and the absolute value of the determinant x1y1z1x2y2z2x3y3z3!!. (5) To see this, first note that the volume of the parallelepiped is given by the product of the area of the base times the height, where the area A of the base is given by equation (3) and the height ˜ U ˜ is just the projection of Z onto the orthogonal complement in ® 3 of the space spanned by X and Y. In other words, if W is the subspace of V = ® 3 spanned by X and Y, then (by Theorem 2.22) V = WÊ • W, and hence by Theorem 2.12 we may write Z = U + aX + bY where U ∞ WÊ and a, b ∞ ® are uniquely determined (the uniqueness of a and b actually follows from Theorem 2.3 together with Theorem 2.12). By definition we have ÓX, UÔ = ÓY, UÔ = 0, and therefore X Y Z U 11.6 VOLUMES IN ®3 587 ÓX,!ZÔ=a˜X˜2+bÓX,!YÔÓY,!ZÔ=aÓY,!XÔ+b˜Y˜2ÓU,!ZÔ=˜U˜2!!. (6) We now wish to solve t he first two of these equations for a and b by Cramer ’s rule (Theorem 4.13). Note that the determinant of the matrix of coefficients is just equation (3), and hence is just the square of the area A of the base of the parallelepiped. Applying Cramer ’s rule we have aA2=ÓX,!ZÔÓX,!YÔÓY,!ZÔÓY,!YÔ=!ÓX,!YÔÓX,!ZÔÓY,!YÔÓY,!ZÔbA2=ÓX,!XÔÓX,!ZÔÓY,!XÔÓY,!ZÔ!!. Denoting the volume by Vol(X, Y, Z), we now have (using the last of equa - tions (6) together with U = Z - aX - bY) Vol 2(X, Y, Z) = A 2 ˜U ˜2 = A 2ÓU, ZÔ = A 2(ÓZ, ZÔ - aÓX, ZÔ - bÓY, ZÔ) so that substituting the expressions for A 2, aA 2 and bA 2, we find Vol2(X,!Y,!Z)=ÓZ,!ZÔÓX,!XÔÓX,!YÔÓY,!XÔÓY,!YÔ+ÓX,!ZÔÓX,!YÔÓX,!ZÔÓY,!YÔÓY,!ZÔ!ÓY,!ZÔÓX,!XÔÓX,!ZÔÓY,!XÔÓY,!ZÔ!!. Using ÓX, YÔ = ÓY, XÔ etc., we see that this is just the expansion of a determi - nant by minors of the third row, and hence (using det A T = det A) Vol2(X,!Y,!Z)=ÓX,!XÔÓY,!XÔÓZ,!XÔÓX,!YÔÓY,!YÔÓZ,!YÔÓX,!ZÔÓY,!ZÔÓZ,!ZÔ=x1x2x3y1y2y3z1z2z3x1y1z1x2y2z2x3y3z3=x1y1z1x2y2z2x3y3z32!!. MULTILINEAR MAPPINGS AND TENSORS 588 We remark that if the collection {X, Y, Z} is linearly dependent, then the volume of the parallelepiped degenerates to zero (since at least one of the paral lelograms that form the sides will have zero area). This agrees with the fact that the determinant (5) will vanish if two rows are linearly dependent. We also note that the area of the base is given by |X!Y|=˜X˜˜Y˜sin!(X,!Y) where the direction of the vector X ª Y is up (in this case). Therefore the pro- jection of Z in the direction of X ª Y is just Z dotted into a unit vector in the direction of X ª Y, and hence the volume of the parallelepiped is given by the number Z  (X ª Y). This is the so-called scalar triple product that should be familiar from elementary courses. We leave it to the reader to show that the scalar triple product is given by the determinant (5) (see Exercise 11.6.1). Finally, note that if any two of the vectors X, Y, Z in equation (5) are interchanged, then the determinant changes sign even though the volume is unaffected (since it must be positive). This observation will form the basis for the concept of “orientation ” to be defined later. Exercises 1. Show that Z  (X ª Y) is given by the determina nt in equation (5). 2. Find the area of the parallelogram whose vertices are: (a) (0, 0), (1, 3), ( -2, 1), and ( -1, 4). (b) (2, 4), (4, 5), (5, 2), and (7, 3). (c) ( -1, 3), (1, 5), (3, 2), and (5, 4). (d) (0, 0, 0), (1, -2, 2), (3, 4, 2), and (4, 2, 4). (e) (2, 2, 1), (3, 0, 6), (4, 1, 5), and (1, 1, 2). 3. Find the volume of the parallelepipeds whose adjacent edges are the vectors: (a) (1, 1, 2), (3, -1, 0), and 5, 2, -1). (b) (1, 1, 0), (1, 0, 1), and (0, 1, 1). 4. Prove both algebraicall y and geometrically that the parallelogram with edges X and Y has the same area as the parallelogram with edges X and Y + aX for any scalar a. 11.6 VOLUMES IN ®3 589 5. Prove both algebraically and geometrically that the volume of the paral - lelepiped in ® 3 with edges X, Y and Z is equal to the volume of the paral - lelepiped with edges X, Y and Z + aX + bY for any scalars a and b. 6. Show that the parallelepiped in ® 3 defined by the three vectors (2, 2, 1), (1, -2, 2) and ( -2, 1, 2) is a cube. Find the volume of this cube. 11.7 VOLUMES IN ®n Now that we have a feeling for volumes in ® 3 expressed as determinants, let us prove the analogous results in ® n. To begin with, we note that parallelo - grams defined by the vectors X and Y in either ® 2 or ® 3 contain all points (i.e., v ectors) of the form aX + b Y for any a, b ∞ [0, 1]. Similarly, given three linearly independent vectors X, Y, Z ∞ ®3, we may define the parallelepiped with these vectors as edges to be that subset of ® 3 containing all vectors of the form aX + bY + cZ where 0 ¯ a, b, c ¯ 1. The corn ers of the parallelepiped are the points ∂1X + ∂ìY + ∂3Z where each ∂á is either 0 or 1. Generalizing these observations, given any r linearly independent vectors Xè, . . . , X r ∞ ®n, we define an r-dimensional paralle lepiped as the set of all vectors of the form a 1Xè + ~ ~ ~ + a r Xr where 0 ¯ aá ¯ 1 for each i = 1, . . . , r. In ®3, by a 1-volume we mean a length, a 2-volume means an area, and a 3- volume is just the usual volume. To define the volume of an r-dimensiona l parallelepiped we proceed by induction on r. In particular, if X is a nonzero vector (i.e., a 1-dimensional parallelepiped) in ® n, we define its 1-volume to be its length ÓX, XÔ 1/2. Proceeding, suppose the (r - 1)-dimensional volume of an (r - 1)-dimensi onal parallelepiped has been defined. If we let P r denote the r-dimensional parallelepiped defined by the r linearly independent vectors Xè, . . . , X r, then we say that the base of P r is the (r - 1)-dimensional paral - lelepiped defined by the r - 1 vectors Xè, . . . , X r-1, and the height of P r is the length of the projection of X r onto the orthogonal complement in ® n of the space spanned by Xè, . . . , X r-1. According to our induction hypothesis, th e volume of an (r - 1)-dimensional parallelepiped has already been defined. Therefore we define the r-volume of P r to be the product of its height times the (r - 1)-dimensional volume of its base. The reader may wonder whet her or not the r-volume of an r-dimensional parallelepiped in any way depends on which of the r vectors is singled out for projection. We proceed as if it does not and then, after the next theorem, we shall show that this is indeed the case. MULTILINEAR MAPPINGS AND TENSORS 590 Theorem 11.13 Let P r be the r-dimensional parallelepiped defined by the r linearly independent vectors Xè, . . . , X r ∞ ®n. Then the r-volume of P r is the positive square root of the determinant ÓX1,!X1ÔÓX1,!X2Ô!ÓX1,!XrÔÓX2,!X1ÔÓX2,!X2Ô!ÓX2,!XrÔ"""ÓXr,!X1ÔÓXr,!X2Ô!ÓXr,!XrÔ!!. (7) Proof For the case of r = 1, we se e that the theorem is true by the definition of length (or 1-volume ) of a vector. Proceeding by induction, we assume the theorem is true for an (r - 1)-dimensional parallelepiped, and we show that it is also true for an r-dimensional parallelepiped. Hence, let us write A2=Vol2(Pr!1)=ÓX1,!X1ÔÓX1,!X2Ô!ÓX1,!Xr!1ÔÓX2,!X1ÔÓX2,!X2Ô!ÓX2,!Xr!1Ô"""ÓXr!1,!X1ÔÓXr!1,!X2Ô!ÓXr!1,!Xr!1Ô for the volume of the (r - 1)-dimensional base of P r. Just as we did in our discussion of volumes in ® 3, we write X r in terms of its projection U onto the orthogonal complement of the space spanned by the r - 1 vectors Xè, . . . , X r. This means that we can write Xr = U + aè Xè + ~ ~ ~ + a r-1Xr-1 where ÓU, XáÔ = 0 for i = 1, . . . , r - 1, and ÓU, X rÔ = ÓU, UÔ. We thus have the system of equations a1ÓX1,!X1Ô+a2ÓX1,!X2Ô+!!!+ar!1ÓX1,!Xr!1Ô=ÓX1,!XrÔa1ÓX2,!X1Ô+a2ÓX2,!X2Ô+!!!+ar!1ÓX2,!Xr!1Ô=ÓX2,!XrÔ""""a1ÓXr-1,!X1Ô+a2ÓXr-1,!X2Ô+!!!+ar!1ÓXr-1,!Xr!1Ô=ÓXr-1,!XrÔ We write Mè, . . . , M r-1 for th e minors of the first r - 1 elements of the last row in (7). Solving the above system for the aá using Cramer ’s rule, we obtain 11.7 VOLUMES IN ®n 591 A2a1=(!1)r!2M1A2a2=(!1)r!3M2!A2ar!1=Mr!1 where the factors of ( -1)r-k-1 in A 2aÉ result from moving the last column of (7) over to become the k th column of the k th minor matrix. Using this result, we now have A2U=A2(!a1X1!a2X2!!!ar!1Xr!1+Xr)=(!1)r!1M1X1+(!1)r!2M2X2+!+(!1)Mr!1Xr!1+A2Xr and hence, using ˜ U ˜2 = ÓU, UÔ = ÓU, X rÔ, we find that (since ( -1)-k = (-1)k) A2˜U˜2=A2ÓU,!XrÔ=(!1)r!1M1ÓXr,!X1Ô+(!1)(!1)r!1M2ÓXr,!X2Ô+!+A2ÓXr,!XrÔ=(!1)r!1[M1ÓXr,!X1Ô!M2ÓXr,!X2Ô+!+(!1)r!1A2ÓXr,!XrÔ]!!. Now note that the right hand side of this equation is precisely the expansion of (7) by minors of the last row, and the left hand side is by definition the square of the r-volume of the r-dimensional parallelepiped P r. This also shows that the deter minant (7) is positive. ˙ This result may also be expressed in terms of the matrix (ÓXá, XéÔ) as Vol(P r) = [det(ÓXá, XéÔ)] 1/2 . The most useful form of this theorem is given in the following corollary. Corollary The n-volume of the n-dimensiona l parallelepiped in ® n defined by the vectors Xè, . . . , Xñ where each Xá has coordinates (x 1i , . . . , x ni) is the absolute value of the determinant of the matrix X given by MULTILINEAR MAPPINGS AND TENSORS 592 X=x11x12!x1nx21x22!x2n"""xn1xn2!xnn!"#####$%&&&&&!!. Proof Note that (det X) 2 = (det X)(det X T) = de t XX T is just the determinant (7) in Theorem 11.13, which is the square of the volume. In other words, Vol(Pñ) = \det X \. ˙ Prior to this theorem, we asked whether or not the r-volume depended on which of the r vectors is singled out for projection. We can now easily show that it does not. Suppose that we have an r-dimensional parallelepiped defined by r linearly independent vectors, and let us label these vectors Xè, . . . , X r. According to Theorem 11.13, we project X r onto the space orthogonal to the space spanned by Xè, . . . , X r-1, and this leads to the d eterminant (7). If we wish to project any other vector instead, then we may simply relabel these r vectors to put a different one into position r. In other words, we have made some permutation of the indices in (7). However, remember that any permuta - tion is a product of transpositions (Theorem 1.2), and hence we need only consider the effect of a single interchange of two indices. Notice, for example, that the indices 1 and r only occur in rows 1 and r as well as in columns 1 and r. And in general, indic es i and j only occur in the i th and j th rows and columns. But we also see that the matrix corresponding to (7) is symmetric about the main diagonal in these indices, and hence an inter- change of the indices i and j has the effect of interchanging both rows i and j as well as columns i and j in exactly the same manner. Thus, because we have interchanged the same rows and columns there will be no sign change, and there fore the deter minant (7) remains unchanged. In particular, it always remains positive. I t now follows that the volume we have defined is indeed indepen dent of which of the r vectors is singled out to be the height of the paral lelepiped . Now note that according to the above corollary, we know that Vol(Pñ) = Vol(Xè, . . . , Xñ) = \det X \ which is always positive. While our discussion just showed that Vol(Xè, . . . , Xñ) is independent of any permutation of indices, the actual value of det X can change sign upon any such permutation. Because of this, we say that the vectors (Xè, . . . , Xñ) are positive ly oriented if det X > 0, and negative ly oriented if det X < 0. Thus the orientation of a set of vectors depends on the order in which they are written. To take into account the sign of det X, we define the oriented volume Vol o(Xè, . . . , Xñ) to be +Vol(Xè, . . . , Xñ) if det X ˘ 0, and -Vol(Xè, . . . , Xñ) if det X < 0. We will return to a careful discussion of orientation in a later section. We also 11.7 VOLUMES IN ®n 593 remark t hat det X is always nonzero as long as the vectors (Xè, . . . , Xñ) are linearly independent. Thus the above corollary may be expressed in the form Vol o(Xè, . . . , Xñ) = det (Xè, . . . , Xñ) where det(Xè, . . . , Xñ) means the determinant as a functio n of the column vectors Xá. Exercises 1. Find the 3-volume of the three -dimensional parallelepipeds in ® 4 defined by the vectors: (a) (2, 1, 0, -1), (3, -1, 5, 2), and (0, 4, -1, 2). (b) (1, 1, 0, 0), (0, 2, 2, 0), and (0, 0, 3, 3). 2. Find the 2-volume of the parallelogram in ® 4 two of whose edges are the vectors (1, 3, -1, 6) and ( -1, 2, 4, 3). 3. Prove that if the vectors X 1, X2, . . . , X r are mutually orthogonal, the r- volume of the parallelepiped defined by them is equal to the product of their lengths. 4. Prove that r vectors X 1, X2, . . . , X r in ® n are linearly dependent if and only if the determinant (7) is equal to zero. 11.8 LINEAR TRANSFORMATIONS AND VOLUMES One of the most useful applications of Theorem 11.13 and its corollary r elates to linear mappings. In fact, this is the approach usually followed in deriving the change of variables formula for multiple integrals. Let {eá} be an ortho - normal basis for ® n, and let Cñ denote the unit cube in ® n. In other words , Cñ = {tèeè + ~ ~ ~ + tñeñ ∞ ®n: 0 ¯ tá ¯ 1} . This is similar to the definition of P r given previously. Now let A: ® n ‘ ® n be a linear transformation. Then the matrix of A relative to the basis {eá} is defined by A(eá) = eéa já. Let us write the image of eá as Xá, so that Xá = A(eá) = eéa já. This means that the column vector Xá has MULTILINEAR MAPPINGS AND TENSORS 594 components (a 1á , . . . , a ná). Under the transformation A, the image of Cñ becomes A(Cñ) = A(Í táeá) = Í táA(eá) = Í táXá (where 0 ¯ tá ¯ 1) which is just the paral lelepiped Pñ spanned by the vectors (Xè, . . . , Xñ). Therefore the volume of Pñ = A(Cñ) is given by \det(Xè , . . . , Xñ) \ = \det (ajá)\ . Recalling that the determinant of a linear transformation is defined to be the determinant of its matrix repres entation, we have proved the next result. Theorem 11.14 Let Cñ be the unit cube in ® n spanned by the orthonormal basis vectors {eá}. If A: ® n ‘ ® n is a linear transformation and Pñ = A(Cñ), then Vol(Pñ) = Vol A(Cñ) = \det A \. It is quite simple to gen eralize this result somewhat to include the image of an n-dimensional parallelepiped under a linear transformation A. First, we note that any parallelepiped Pñ is just the image of Cñ under some linear transforma tion B. Indeed, if Pñ = {t 1Xè + ~ ~ ~ + tñ Xñ: 0 ¯ t á ¯ 1} for some set of vectors Xá, then we may define the transformation B by B(eá) = Xá, and hence Pñ = B(Cñ). Thus A(Pñ) = A(B(Cñ)) = (A ı B)(Cñ) and therefore (using Theorem 11.14 along with the fact that the matrix of the composition of two transforma tions is the matrix product) VolA(Pn)=Vol[(A!B)(Cn)]=|det(A!B)|=|detA||detB|=|detA|Vol(Pn)!!. In other words, \det A \ is a measure of how much the volume of the parallel - epiped changes under the linear transformation A. See the figure below for a picture of this in ® 2. We s ummarize this discussion as a corollary to Theorem 11.14. Corollary Suppose Pñ is an n-dimensional parallelepiped in ® n, and let A: ®n ‘ ® n be a linear transformation. Then Vol A(Pñ) = \det A \Vol(Pñ). 11.8 LINEAR TRANSFORMATIONS AND VOLUMES 595 Now that we have an intuitive grasp of these concepts, let us look at this material from the point of view of exterior algebra. This more sophisticated approach is of great use in the theory of integration. Let U and V be real vector spaces. Recall from Theorem 9.7 that given a linear trans formation T ∞ L(U, V), we defined the transpose mapping T* ∞ L(V*, U*) by T*ø = ø ı T for all ø ∞ V*. By this we mean if u ∞ U, then T*ø(u) = ø(Tu). As we then saw in Theorem 9.8, if U and V are finite -dimensional and A is the matrix representation of T, then A T is the matrix representation of T*, and hence certain properties of T* follow naturally. For example, if Tè ∞ L(V, W) and Tì ∞ L(U, V), then (Tè ı Tì)* = Tì* ı Tè* (Theorem 3.18), and if T is nonsingular, then (Tî)* = (T*)î (Corollary 4 of Theorem 3.21). Now suppose that {eá} is a basis for U and {fé} is a basis for V. To keep the notation simple and understandable, let us write the corresponding dual bases as {e i} and {f j}. We define the matrix A = (a ji) of T by Teá = féa já. Then (just as in the proof of T heorem 9.8) (T*fi)ej=fi(Tej)=fi(fkakj)=akjfi(fk)=akj!ik=aij=aik!kj=aikek(ej) which shows that T*fi=aikek!!. (8) We will use this result frequently below. We now generalize our definition of the transpose. If ƒ ∞ L(U, V) and T ∞ Tr (V), we define the pull-back ƒ* ∞ L(Tr (V), Tr (U)) by (ƒ*T )(uè, . . . , u r) = T(ƒ(uè), . . . , ƒ(u r)) A B eè eì Cì Xè Xì Xè = B(e è) Xì = B(e ì) Pì A(Xì) A(Xè) A(Pì) MULTILINEAR MAPPINGS AND TENSORS 596 where uè, . . . , u r ∞ U. Note that in the particula r case of r = 1, the mapping ƒ* is just the transpose of ƒ. It should also be clear from the definition that ƒ* is indeed a linear transformation, and hence ƒ*(aTè + bTì) = aƒ*Tè + bƒ*Tì . We also emphasize that ƒ need not be an isomorphism for us to define ƒ*. The main properties of the pull-back are given in the next theorem. Theorem 11.15 If ƒ ∞ L(U, V) and √ ∞ L(V, W), then (a) (√ ı ƒ)* = ƒ* ı √*. (b) If I ∞ L(U) is the identity map, then I* is the identity in L( Tr(U)). (c) If ƒ is an isomorphism, then so is ƒ*, and (ƒ*)î = (ƒî)*. (d) If Tè ∞ Trè(V) and Tì ∞ Trì(V), then ƒ*(Tè · Tì) = (ƒ*Tè) · (ƒ*Tì) . (e) Let U have basis {eè, . . . , e m}, V have basis {fè, . . . , fñ} and suppose that ƒ(eá) = féa já. If T ∞ Tr (V) has components T iè ~ ~ ~ i‹ = T(f iè , . . . , f i‹), then the components of ƒ*T relative to the basis {eá} ar e given by (ƒ*T )jè~ ~ ~j‹ = T iè~ ~ ~i‹ aièjè ~ ~ ~ a i‹j‹ . Proof (a) Note that √ ı ƒ: U ‘ W, and hence ( √ ı ƒ)*: Tr (W) ‘ Tr (U). Thus for any T ∞ Tr (W) and uè, . . . , u r ∞ U we have ((!!")*T)(u1,!…!,ur)=T(!("(u1)),!…!,!!("(ur)))=(!*T)("(u1),!…!,!"(ur))=(("*!!*)T)(u1,!…!,!ur)!!. (b) Obvious from the definition of I*. (c) If ƒ is an isomorphism, then ƒî exists and we have (using (a) and (b)) ƒ* ı (ƒî)* = (ƒî ı ƒ)* = I* . Similarly (ƒî)* ı ƒ* = I*. Hence (ƒ*)î exists and is equal to (ƒî)*. (d) This follows directly from the definitions (see Exercise 11.8.1). (e) Using the definitions, we have 11.8 LINEAR TRANSFORMATIONS AND VOLUMES 597 (!*T)j1!jr=(!*T)(ej1,!…,!ejr)=T(!(ej1),…,!!(ejr))=T(fi1ai1j1,…,!firairjr)=T(fi1,…,!fir)ai1j1!airjr=Ti1!irai1j1!airjr!!. Alternatively, if {e i}and {f j} are the bases dual to {eá} and {fé} respec - tively , then T = T iè~ ~ ~i‹ eiè · ~ ~ ~ · e i‹ and consequently (using the linearity of ƒ*, part (d) and equation (8)), !*T=Ti1!ir!*ei1"!"!*eir=Ti1!irai1j1!airjrfj1"!"fjr which therefore yields the same result. ˙ For our present purposes, we will only need to consider the pull-back as defined on the space „ r(V) rather than on Tr (V). Therefore, if ƒ ∞ L(U, V) then ƒ* ∞ L(Tr (V), Tr (U)), and hence we see that for ø ∞ „r(V) we have (ƒ*ø)(uè, . . . , u r) = ø(ƒ(uè), . . . , ƒ(u r)). This shows that ƒ*(„ r(V)) ™ „ r(U). Parts (d) and (e) of Theorem 11.15 applied to the space „ r(V) yield the fol- lowing special cases. (Recall that \i1 ~ ~ ~ i r\ means the sum is over increasing indices iè < ~ ~ ~ < i r.) Theorem 11.16 Suppose ƒ ∞ L(U, V), å ∞ „r(V) and ∫ ∞ „s(V). Then (a) ƒ*(å° ∫) = (ƒ*å)°(ƒ*∫ ). (b) Let U and V have bases {eá} and {fá} respectively, and let U* and V* have bases {e i} and {f i}. If we write ƒ(eá) = féa já and ƒ*(f i ) = a iéej, and if å = a\ iè~ ~ ~i‹\ f iè°~ ~ ~°f i‹ ∞ „r(V), then ƒ*å = aÄ \ kè~ ~ ~k‹ \ ekè° ~ ~ ~ °ek‹ where ˆa|k1!kr|=a|i1!ir|!k1!krj1!jrai1j1!airjr!!. Thus we may write aÄkè~ ~ ~ k‹ = a \ iè~ ~ ~i‹ \ det(a I K) where MULTILINEAR MAPPINGS AND TENSORS 598 det(aIK)=ai1k1!ai1kr""airk1!airkr!!. Proof (a) For simplicity, let us write (ƒ*å)(u J) = å(ƒ(u J)) instead of (ƒ*å)(uè, . . . , u r) = å(ƒ(uè), . . . , ƒ(u r)) (see the discussion following Theorem 11.4). Then, in an obvious notation, we have [!*("#$)](uI)=("#$)(!(uI))=%!J,!K&IJK("(!(uJ))$(!(uK))=%!J,!K&IJK(!*")(uJ)(!*$)(uK)=[(!*")#(!*$)](uI)!!. By induction, this also obviously applies to the wedge product of a fini te number of forms. (b) From å = a \ iè~ ~ ~i‹ \ f iè° ~ ~ ~ °f i‹ and ƒ*(f i) = a i é ej, we have (using part (a) and the linearity of ƒ*) !*"=a|i1!ir|!*(fi1)#!#!*(fir)=a|i1!ir|ai1j1!airjrej1#!#ejr!!. But ej1!!!ejr=""K#k1!krj1!jrek1!!!ekr and hence we have !*"=a|i1!ir|#"K$k1!krj1!jrai1j1!airjrek1%!%ekr=ˆa|k1!kr|ek1%!%ekr where ˆa|k1!kr|=a|i1!ir|!k1!krj1!jrai1j1!airjr!!. Finally, from the definition of determinant we see that !k1!krj1!jrai1j1!airjr=ai1k1!ai1kr""airk1!airkr!!.!!˙ 11.8 LINEAR TRANSFORMATIONS AND VOLUMES 599 Example 11.11 (This is a continuation of Example 11.1.) An important example of ƒ*å is related to the change of variables formula in multipl e integrals. While we are not in any position to pre sent this material in detail, the idea is this. Suppose we consider the spaces U = ® 3(u, v, w) and V = ®3(x, y, z) where the letters in parentheses tell us the coordinate system used for that particular copy of ® 3. Note that if we write (x, y, z) = (x 1, x2, x3) and (u, v, w) = (u 1, u2, u3), then from elementary calculus we know that dx i = ($xi/$uj)duj and $/$u i = ($x j/$ui)($/$x j). Now recall from Example 11.1 that at each point of ® 3(u, v, w), the tan- gent space has the basis {eá} = {$/$u i} and the cotangent space has the corre - sponding dual basis {e i} = {du i}, with a similar result for ® 3(x, y, z). Let us define ƒ: ® 3(u, v, w) ‘ ® 3(x, y, z) by ƒ($/$u i) = ($x j/$ui)($/$x j) = a já($/$x j) . It is then apparent that (see equation (8)) ƒ*(dx i) = a iéduj = ($x i/$uj)duj as we should have expected. We now apply this to the 3-form å = a 123 dx1°dx 2°dx 3 = dx°dy°dz ∞ „ 3(V) . Since we are dealing with a 3-form in a 3-dimensional space, we must have ƒ*å = aÄ du°dv°dw where aÄ = aÄ 123 consists of the single term given by the determinant a11a12a13a21a22a23a31a32a33=!x1/!u1!x1/!u2!x1/!u3!x2/!u1!x2/!u2!x2/!u3!x3/!u1!x3/!u2!x3/!u3 which the reader may recognize as the so-called Jacobian of the transforma - tion. This determinant is usually writt en as $(x, y, z)/$(u, v, w), and hence we see that !*(dx"dy"dz)=#(x,!y,!z)#(u,!v,!w)du"dv"dw!!. MULTILINEAR MAPPINGS AND TENSORS 600 This is precisely how volume elements transform (at least locally), and hence we have formulated the change of variables formula in quit e general terms. ∆ This formalism allows us to define the determinant of a linear transforma - tion in an interesting abstract manner. To see this, suppose ƒ ∞ L(V) where dim V = n. Since dim „ n(V) = 1, we may choose any nonzero øà ∞ „n(V) as a basis. The n ƒ*: „ n(V) ‘ „ n(V) is linear, and hence for any ø = càøà ∞ „n(V) we have ƒ*ø = ƒ*(càøà) = càƒ*øà = càcøà = c(càøà) = cø for some scalar c (since ƒ*øà ∞ „n(V) is necessarily of the form cøà). Noting that this result did not depend on the scalar cà and hence is independent of ø = càøà, we see that the scalar c must be unique. We therefore define the deter - minant of ƒ to be the unique scalar, denoted by det ƒ, such that ƒ*ø = (det ƒ)ø . It is important to realize tha t this definition of the determinant does not depend on any choice of basis for V. However, let {eá} be a basis for V, and define the matrix (a ié) of ƒ by ƒ(eá) = eéa já. Then for any nonzero ø ∞ „n(V) we have (ƒ*ø)(eè, . . . , eñ) = (det ƒ)ø(eè, . . . , eñ) . On the other hand, Example 11.2 shows us that (!*")(e1,!…!,!en)="(!(e1),!…!,!!(en))=ai11!ainn"(ei1,!…!,ein)=(det(aij))"(e1,!…!,en)!!. Since ø ≠ 0, we have therefore proved the next result. Theorem 11.17 If V has basis {eè, . . . , eñ} and ƒ ∞ L(V) has the matrix representation (a ié) defined by ƒ(eá) = eéa já, then det ƒ = det(a ié). In other words, our abstract definition of the determinant is exactly the same as our earlier classical definition. In fact, it is no w easy to derive some of the properties of the determinant that were not exactly simple to prove in the more traditional manner. 11.8 LINEAR TRANSFORMATIONS AND VOLUMES 601 Theorem 11.18 If V is finite -dimensional and ƒ, √ ∞ L(V, V), then (a) det(ƒ ı √) = (det ƒ)(det √). (b) If ƒ is the identity transformation, then det ƒ = 1. (c) ƒ is an isomorphism if and only if det ƒ ≠ 0, and if this is the case, then det ƒî = (det ƒ)î. Proof (a) By definition we have (ƒ ı √)*ø = det(ƒ ı √)ø. On the other hand, by Theorem 11.15(a) we know that (ƒ ı √)* = √* ı ƒ*, and hence (!!")*#="*(!*#)="*[(det!)#]=(det!)"*#=(det!)(det")#!!. (b) If ƒ = 1 then ƒ* = 1 also (by Theorem 11.15(b)), and thus ø = ƒ*ø = (det ƒ)ø implies det ƒ = 1. (c) First assume that ƒ is an isomorphism so that ƒî exists. Then by parts (a) and (b) we see that 1 = det(ƒƒî) = (det ƒ)(det ƒî) which implies det ƒ ≠ 0 and det ƒî = (det ƒ)î. Conversely, suppose that ƒ is not an isomorphism. Then Ker ƒ ≠ 0 and there exists a nonzero eè ∞ V such that ƒ(eè) = 0. By Theorem 2.10, we can extend this to a basis {eè, . . . , eñ} for V. But then for any nonzero ø ∞ „n(V) we have (det!)"(e1,!…!,en)=(!*")(e1,!…!,en)="(!(e1),!…!,!(en))="(0,!!(e2),!…!,!!(en))=0 and hence we must have det ƒ = 0. ˙ Exercises 1. Prove Theorem 11.15(d). 2. Show that the matrix ƒ*T defined in Theorem 11.15(e) is just the r-fold Kronecker produc t A · ~ ~ ~ · A where A = (a ij). MULTILINEAR MAPPINGS AND TENSORS 602 The next three exercises are related. 3. Let ƒ ∞ L(U, V) be an isomorphism, and suppose T ∞ Tr Í(U). Define the push -forward ƒ§ ∞ L(Tr Í(U), Tr Í(V)) by ƒ§T(å 1, . . . , å s, u1, . . . , u r) = T( ƒ*å 1, . . . , ƒ*å s, ƒîu 1, . . . , ƒîu r) where å 1, . . . , å s ∞ U* and u 1, . . . , u r ∞ U. If √ ∞ L(V, W) is also an iso- morphism , prove the following: (a) (√ ı ƒ)§ = √§ ı ƒ§. (b) If I ∞ L(U) is the identity map, then so is I§ ∞ L(Tr Í(U)). (c) ƒ§ i s an isomorphism, and (ƒ§)î = (ƒî)§. (d) If T 1 ∞ Tr1s1(U) and T 2 ∞ Tr2s2(U) , then ƒ§(T 1 · T 2) = (ƒ§T 1) · (ƒ§T 2) . 4. Let ƒ ∞ L(U, V) be an isomorphism, and let U and V have bases {e i} and {fi} respectively. Defin e the matrices (a ij) and (b ij) by ƒ(e i) = f jaji and ƒî(f i) = e jbji. Suppose T ∞ Tr Í(U) has components T iè ~ ~ ~ i› jè ~ ~ ~ j‹ relative to {e i}, and S ∞ Tr Í(V) has components S iè ~ ~ ~ i› jè ~ ~ ~ j‹ relative to {f i}. Show that the components of ƒ§T a nd ƒ§S are given by (!§T)i1!isj1!jr=ai1p1!aispsTp1!psq1!qrbq1j1!bqrjr(!§S)i1!isj1!jr=bi1p1!bispsSp1!psq1!qraq1j1!aqrjr!!. 5. Let {ø i} be the basis dual to {e i} for ® 2. Let T = 2e 1 · ø 1 - e2 · ø 1 + 3e 1 · ø 2 and suppose ƒ ∞ L(® 2) and √ ∞ L(® 3, ®2) have the matrix representations !=!!21"11#$%&'( and )=0!1"11!0!!2#$%&'(!!. Compute Tr T, ƒ*T, √*T, Tr( √*T), and ƒ§T. 11.9 ORIENTATIONS AND VOL UMES 603 11.9 ORIENTATIONS AND VOLUMES Suppose dim V = n and consider the space „ n(V). Since this space is 1- dimen sional , we consider the n-form ø = e 1° ~ ~ ~ °en ∞ „ n(V) where the basis {e i} for V* is dual to the basis { eá} for V. If {vá = e jvji} is any set of n linearly independent vectors in V then, according to Examples 11.2 and 11.8, we have ø(vè, . . . , vñ) = det(v ji)ø(eè, . . . , eñ) = det(v ji) . However, from the corollary to Theorem 11.13, this is just the oriented n- volume of the n-dimensional parallelepiped in ® n spanned by the vectors {vá}. Therefore, we see that an n-form in some sense represents volumes in an n- dimensional space. We now proceed to make this definition precise, begin ning with a car eful definition of the notion of orientation on a vector space. In order to try and make the basic idea clear, let us first consider the space ®2 with all possible orthogonal coordinate systems. For example, we may consider the usual “right -handed” coord inate system {eè, eì} shown below, or we may consider the alternative “left-handed” system {eæè, eæì} also shown. In the first case, we see that rotating eè into eì through the smallest angle between them involves a counterclockwise rotation, while in the second case, rotating eæè into eæì entails a clockwise rotation. This effect is shown in the elementary vector cross product, where the direction of eè ª eì is defined by the “right -hand rule” to point out of the page, while eæè ª eæì points into the page. We now ask whether or not it is possible to continuously rotate eæè into eè and eæì into eì while maintaining a basis at all times. In other words, we ask if these two bases are in some sense equivalent. Without being rigorous, it should be clea r that this can not be done because there will always be one point where the vectors eæè and eæì will be co-linear , and hence linearly dependent. This observation suggests that we consider the determinant of the matrix representing this change of basis. eè eì eæì eæè MULTILINEAR MAPPINGS AND TENSORS 604 In order to formulate this idea precisely, let us take a look at the matrix relating our two bases {eá} and {eæá} for ® 2. We thus write eæá = eéa já and investigate the determinant det(a ié). From the above figure, we see that !e1=e1a11+e2a21where a11<0 and a21>0!e2=e1a12+e2a22where a12<0 and a22>0 and hence det(a ié) = a 1è a2ì - a1ì a2è < 0. Now suppose that we view this transformation as a continuous modifica - tion of the identity transformation. This means we consider the basis vectors eæá to be continuous functions eæá(t) of the matrix a já(t) for 0 ¯ t ¯ 1 where a já(0) = ∂já and a já(1) = a já, so that eæá(0) = eá and eæá(1) = eæá. In other words, we write eæá(t) = eéa já(t) for 0 ¯ t ¯ 1. Now note that det(a ié(0)) = det( ∂ié) = 1 > 0, while det(a ié(1)) = det(a ié) < 0. Therefore, since the determinant is a continuous function of its entries, there must be some value tà ∞ (0, 1) where det(a ié(tà)) = 0. It then follows that the vectors eæá(tà) will be linearly depen dent. What we have just shown is that if we start with any pair of linearly inde- pendent vectors, and then tr ansform this pair into another pair of linearly independent vectors by moving along any continuous path of linear transfor - mations that always maintains the linear independence of the pair, then every linear transformation along this path must have positi ve determinant. Another way of saying this is that if we have two bases that are related by a transfor - mation with negative determinant, then it is impossible to continuously trans - form one into the other while maintaining their independence. This argument clearly applies to ® n and is not restricted to ® 2. Conversely, suppose we had assumed that eæá = eéa já, but this time with det(a ié) > 0. We want to show that {eá} may be continuously transformed into {eæá} while maintaining linear independence all the wa y. We first assume that both {eá} and {eæá} are orthonormal bases. After treating this special case, we will show how to take care of arbitrary bases. (Unfortunately, the argument we are about to give relies on the topological concept of path connectedne ss. Since a complete discussion of this topic would lead us much too far astray, we shall be content to present only the fundamental concepts in Appendix C. Besides, this discussion is only motiva - tion, and the reader should not get too bogged down in the details of this argument. Those readers who know some topology should have no trouble filling in the neces sary details if desired.) Since {eá} and {eæá} are orthonormal, it follows from Theorem 10.6 (applied to ® rather than ç) that the transformation ma trix A = (a ié) defined by eæá = eéa já must be orthogonal, and hence det A = +1 (by Theorem 10.8(a) and 11.9 ORIENTATIONS AND VOL UMES 605 the fact that we are assuming {eá} and {eæá} are related by a transformation with positive determinant). By Theorem 10.19, there exists a nonsingular matrix S such that SîAS = M œ where M œ is the block diagonal canonical form consist ing of +1 ’s, -1’s, and 2 x 2 rotation matrices R(œá) given by R(!i)=cos!i"sin!isin!i!!cos!i#$%&'(!!. It is important to realize that if there are more than two +1 ’s or more than two -1’s, then each pair may be combined into one of the R(œá) by choosing either œá = π (for each pair of -1’s) or œá = 0 (for each pair of +1 ’s). In this manner, we view M œ as consisting entirely of 2 x 2 rotation matrices, and at most a single +1 and/or -1. Since det R(œá) = +1 for any œá, we see that (using Theorem 4.14) det Mœ = +1 if there is no -1, and det M œ = -1 if there is a single -1. From A = SM œSî, we see that det A = det M œ, and since we are requiring that det A > 0, we must have the case where ther e is no -1 in M œ. Since cos œá and sin œá are continuous functions o f œá ∞ [0, 2π ) (where the interval [0, 2π ) is a path connected set), we note that by parametrizing each œá by œá(t) = (1 - t)œá, the matrix M œ may be continuously connected to the identity matrix I (i.e., at t = 1). In other words, we consider the matrix M œ(t) where M œ(0) = M œ and M œ(1) = I. Hence every such M œ (i.e., any matrix of the same form as our particular M œ, but with a different set of œá ’s) may be continuously connected to the identity matrix. (For those readers who know some topology, note all we have said is th at the torus [0, 2π ) ª ~ ~ ~ ª [0, 2π ) is path connected, and hence so is its continuous image which is the set of all such M œ.) We may write the (infinite) collection of all such M œ as M = {M œ}. Clearly M is a path connected set. Since A = SM œSî and I = SISî, we see that both A and I are contained in the collection SMSî = {SM œSî}. But SMSî is also path connected since it is just the continuous image of a path connected set (matrix multiplication is obviously continuous). Thus we have shown that both A and I lie in the path connected set SMSî, and hence A may be continuously connected to I. Note also that every transformation along this path has positive determinant since det SM œSî = det M œ = 1 > 0 for every Mœ ∞ M. If we now take any path in SMSî that starts at I and goes to A, then applying this path to the basis {eá} we obtain a continuous transformation from {eá} to {eæá} with everywhere positive determinant. This completes the proof for the special case of orthonormal bases. Now suppose that {vá} and {væá} are arbitrary bases related by a transfor - mation with positive determinant. Starting with the basis {vá}, we first apply MULTILINEAR MAPPINGS AND TENSORS 606 the Gram -Schmidt process (Theorem 2.21) to {vá} to obtain an orthonormal basis {eá} = {véb já}. This orthonormalization process may be visualized as a sequence vá(t) = véb já(t) (for 0 ¯ t ¯ 1) of continuous scalings and rotations that always maintain linear independence such that vá(0) = vá (i.e., b já(0) = ∂já) and vá(1) = eá (i.e., b já(1) = b já). Hence we have a continuous transformation b já(t) taking {vá} into {eá} with det(b já(t)) > 0 (the transformation starts with det(b já(0)) = det I > 0, and since the vectors are always independent, it must main tain det((b já(t)) ≠ 0). Similarly, we may transform {væá} into an orthonor - mal basis {eæá} by a continuous transformation with positive determinant. (Alternatively, it was shown in Exercise 5.4.14 that the Gram -Schmidt process is represented by an upper -triangular matrix with all positive diagonal ele- ments , and hence its determinant is positive.) Now {eá} and {eæá} are related by an orthogonal transformation that must also have determinant equal to +1 because {vá} and {væá} are related by a transformation with positive determi - nant, and both of the Gram -Schmidt transformations have positive determi - nant. This reduces the general case to the special case treated above. With this discussion as motivation, we make the following definition. Let {vè, . . . , vñ } and {væè, . . . , væñ} be two ordered bases for a real vector space V, and assume that væá = véa já. These two bases are said to be similarly oriented if det(a ié) > 0, and we write this as {vá} — {væá}. In other words, {vá} — {væá} if væá = ƒ(vá) with det ƒ > 0. We leave it to the reader to show that this defines an equivalence relation on the set of all ordered bases for V (see Exercise 11.9.1). We denote the equivalence class of the basis {vá} by [vá]. It is worth point ing out that had we instead required det(a ié) < 0, then this would not have defined an equivalence relation. This is because if (b ié) is another such transformation with det(b ié) < 0, then det(a iébjÉ) = det(a ié)det(b jÉ) > 0 . Intuitively this is qui te reasonable since a combination of two reflections (each of which has negative determinant) is not another reflec tion. We now define an orientation of V to be an equivalence class of ordered bases. The space V together with an orie ntation [vá] is called an oriented vector space (V, [vá]). Since the determinant of a linear transformation that relates any two bases must be either positive or negative, we see that V has exactly two orientations. In partic ular, if {vá} is any given basis, then every other basis belonging to the equivalence class [vá] of {vá} will be related to {vá} by a transformation with positive determinant, while those bases related to {vá} by a transformation with negative determinant will be related to each other by a transformation with positive determinant (see Exercise 11.9.1). 11.9 ORIENTATIONS AND VOL UMES 607 Now recall we have seen that n-forms seem to be related to n-volumes in an n-dimensional space V. To precisely define this relationship, we formulate orient ations in terms of n-forms . To begin with, the nonzero elements of the 1- dimensional space „ n(V) are called volume forms (or sometimes volume elements ) on V. If øè and øì are volume forms, then øè is said to be equiv - alent to øì if øè = cøì for some real c > 0, and in this case we also write øè — øì. Since every element of „ n(V) is related to every other element by a rela- tionship of the form øè = aøì for some real a (i.e., -Ÿ < a < Ÿ), it is clear tha t this equivalence relation divides the set of all nonzero volume forms into two distinct groups (i.e., equivalence classes). We can relate any ordered basis {vá} for V to a specific volume form by defining ø = v 1° ~ ~ ~ °vn where {v i} is the basis dua l to {vá}. That this association is meaningful is shown in the next result. Theorem 11.19 Let {vá} and {võá} be bases for V, and let {v i} and {võ i} be the corresponding dual bases. Define the volume forms ø = v 1° ~ ~ ~ °vn and øù = võ1° ~ ~ ~ °võn . Then {vá} — {võá} if and only if ø — øù. Proof First suppose that {vá} — {võá}. Then võá = ƒ(vá) where det ƒ > 0, and hence (using ø(vè , . . . , vñ) = v 1° ~ ~ ~ °vn (vè , . . . , vñ) = 1 as shown in Example 11.8) !(v1,!…!,vn)=!("(v1),!…!,"(vn))=("*!)(v1,!…!,vn)=(det")!(v1,!…!,vn)=det"!!. If we assume that ø = cøù for some -Ÿ < c < Ÿ, then using øù(võè , . . . , võñ) = 1 we see that our result implies c = det ƒ > 0 and thus ø — øù. MULTILINEAR MAPPINGS AND TENSORS 608 Conversely, if ø = c øù where c > 0, then assuming that võá = ƒ(vá), the above calculation shows that det ƒ = c > 0, and henc e {vá} — { võá}. ˙ What this theorem shows us is that an equivalence class of bases uniquely determines an equivalence class of volume forms and conversely. Therefore it is consistent with our earlier definitions to say that an equivalence class [ø] of volume forms on V defines an orientation on V, and the space V together with an orientation [ø] is called an oriented vector space (V, [ø]). A basis {vá} for (V, [ø]) is now said to be positively oriented if ø(vè, . . . , vñ) > 0. Not surprisingly, the equivalence class [ -ø] is called the reverse orientation , and the basis {vá} is said to be negatively oriented if ø(vè, . . . , vñ) < 0. Note that if the ordered basis {vè, vì , . . . , vñ} is negatively oriented, then the basis {vì, vè , . . . , vñ} will be positively oriented because ø(vì, vè , . . . , vñ) = - ø(vè , vì , . . . , vñ) > 0. By way of additional terminology, the standard orientation on ® n is that orientation defined by either the standard ordered basis {eè, . . . , eñ}, or the corresponding volume form e 1° ~ ~ ~ °en. In order to proceed any further, we must introduce the notion of a metric on V. This is the subject of the next section. Exercises 1. (a) Show that the collection of all similarly oriented bases for V defines an equivalence relation on the set of all ordered bases for V. (b) Let {vá} be a basis for V. Show that all other bases related to {vá} by a transformation with negative determinant will be related to each other by a transformation with positive determinant. 2. Let (U, ø) and (V, µ) be oriented vector spaces with chosen volume ele- ments . We say that ƒ ∞ L(U, V) is volume preserving if ƒ*µ = ø. If dim U = dim V is finite, show that ƒ is an isomorphism. 3. Let (U, [ø]) and (V, [µ]) be oriented vector spaces. We say that ƒ ∞ L(U, V) is orien tation preserving if ƒ*µ ∞ [ø]. If dim U = dim V is finite, show that ƒ is an isomorphism. If U = V = ® 3, give an example of a linear transformation that is orientation preserving but not volume preserv ing. 11.10 THE METRIC TENSOR AN D VOLUME FORMS 609 11.10 THE METRIC TENSOR AND VOLUME FORMS We now generalize slightly our definition of inner products on V. In particu - lar, recall from Section 2.4 (and the beginning of Section 9.2) that property (IP3) of an inner product requires that Óu, uÔ ˘ 0 for all u ∞ V and Óu, uÔ = 0 if and only if u = 0. If we drop this condition entirely, then we obtain an indefinite inner product on V. (In fact, some authors define an inner product as obeying only (IP1) and (IP2), a nd then refer to what we have called an inner product as a “positive definite inner product. ”) If we replace (IP3) by the weaker require ment (IP3æ) Óu, vÔ = 0 for all v ∞ V if and only if u = 0 then our inner product is said to be nondegenerate . (Note that every example of an inner product given in this book up to now has been nondegenerate.) Thus a real nondegenerate indefinite inner product is just a real nondegenerate symmetric bilinear map. We will soon see an e xample of an inner product with the property that Óu, uÔ = 0 for some u ≠ 0 (see Example 11.13 below). Throughout the remainder of this chapter, we will assume that our inner products are indefinite and nondegenerate unless otherwise noted. We further - more assume that we are dealing exclusively with real vector spaces. Let {eá} be a basis for an inner product space V. Since in general we will not have Óeá, eéÔ = ∂áé, we define the scalars gáé by gáé = Óeá, eé Ô . In terms of the gáé, we have for a ny X, Y ∞ V ÓX, YÔ = Óx ieá, y jeéÔ = x i yjÓeá, eéÔ = gáéx iyj . If {eõá} is another basis for V, then we will have eõá = eéa já for some nonsingu lar transition matrix A = (a já). Hence, writing gõáé = Óeõá, eõéÔ we see that gõáé = Óeõá, eõéÔ = Óerari, esasjÔ = a riasjÓer, esÔ = a riasjgrs which shows that the gáé transform like the components of a second -rank covariant tensor. Indeed, defining the tensor g ∞ T2 (V) by g(X, Y) = ÓX, YÔ results in g(eá, eé) = Óeá, eéÔ = gáé MULTILINEAR MAPPINGS AND TENSORS 610 as it sho uld. We are therefore justified in defining the ( covariant ) metric tensor g = gáéø i · ø j ∞ T 2 (V) (where {ø i} is the basis dual to {eá}) by g(X, Y) = ÓX, YÔ. In fact, since the inner product is nondegenerate and symmet ric (i.e., ÓX, YÔ = ÓY, XÔ), we see that g is a nondegenerate symmetric tensor (i.e., gáé = géá). Next, we notice that given any vector A ∞ V, we may define a linear func- tional ÓA, Ô on V by the assignment B ’ ÓA, BÔ. In other words, for any A ∞ V, we a ssociate the 1-form å defined by å(B) = ÓA, BÔ for every B ∞ V. Note that the kernel of the mapping A ’ ÓA, Ô (which is easily seen to be a vector space homomorphism) consists of only the zero vector (since ÓA, BÔ = 0 for every B ∞ V implies that A = 0), a nd hence this association is an iso - morphism. Given any basis {eá} for V, the components aá of å ∞ V* are given in terms of those of A = a ieá ∞ V by aá = å(eá) = ÓA, eáÔ = Óa j eé, eáÔ = a jÓeé, eáÔ = a jgéá Thus, to any contravariant vector A = ai eá ∞ V, we can associate a unique covariant vector å ∞ V* by å = aáø i = (a jgéá)ø i where {ø i} is the basis for V* dual to the basis {eá} for V. In other words, we write aá = a jgéá and we say that aá arises by lowering the index j of a j. Example 11.12 If we consider the space ® n with a Cartesian coordinate sys- tem {eá}, then we have gáé = Óeá, eé Ô = ∂áé, and hence aá = ∂áéaj = a i. Therefore, in a Cartesian coordinate system , there is no distinction between the comp o- nents of covariant and contravariant vectors. This explains why 1-forms never arise in elementary treatments of vector analysis. ∆ Since the metric tensor is nondegenerate, the matrix (gáé) must be nonsin - gular (or else the mapping a j ’ aá would not b e an isomorphism). We can there fore define the inverse matrix (g ij ) by gijgéÉ = gÉég ji = ∂iÉ . 11.10 THE METRIC TENSOR AN D VOLUME FORMS 611 Using (g ij ), we see that the inverse of the mapping a j ’ aá is given by gijaé = a i . This is called, naturally enough, raising an index . We will show below that the g ij do indeed form the components of a tensor. It is worth remarking that the “tensor ” gié = g ikgÉé = ∂ié (= ∂éi ) is unique in that it has the same components in any coordinate system. Indeed, if {eá} an d {eõá} are two bases for a space V with corresponding dual bases {ø i} and {øù i}, then eõá = eéa já and øù j = b jáøi = (aî) jáøi (see the discussion following Theorem 11.2). Therefore, if we define the tensor ∂ to have the same values in the first coordinate system as the Kronecker delta, then ∂ié = ∂(øi, eé). If we now define the symbol ∂äié by ∂äié = ∂(øùi, eõé), then we see that !ij=!("i,!ej)=!((a#1)ik"k,!erarj)=(a#1)ikarj!("k,!er)=(a#1)ikarj!kr=(a#1)ikakj=!ij!!. This shows that the ∂ié are in fact the components of a tensor, and that these components are the sa me in any coordinate system. We would now like to show that the scalars g ij are indeed the components of a tensor. There are several ways that this can be done. First, let us write gáégjk = ∂ki where we know that both gáé and ∂ki are tensors. Multiplying both sides of this equation by (aî) rÉais and using (aî) rÉais∂ki = ∂rs we find gáég jk(a î)rÉais = ∂rs . Now substitute gáé = gá t∂té = gitatq(aî) qé to obtain [a is atq git][(aî) qé(aî) rÉgjk] = ∂rs . Since g it is a tensor, we know that a isatq git = gõsq. If we write gõq r = (aî) qé(aî) rÉgjk then we will have defined the g jk to transform as the components of a tensor, and furthermore, they have the requisite property that gõsq gõqr = ∂rs. Therefore we have defined the ( contravariant ) metric te nsor G ∞ T 0 ™(V) by MULTILINEAR MAPPINGS AND TENSORS 612 G = g ijeá · eé where g ijgéÉ = ∂iÉ. There is another interesting way for us to define the tensor G. We have already seen that a vector A = a i eá ∞ V defines a unique linear form å = aéøj ∞ V* by the association å = gáéa iøj. If we denote the inverse of the matrix (gáé) by (g ij) so that g ijgéÉ = ∂iÉ, then to any linear form å = aáø i ∞ V* there corresponds a unique vector A = a i eá ∞ V defined by A = g ijaáeé. We can now use this isomorphism to defi ne an inner product on V*. In other words, if Ó , Ô is an inner product on V, we define an inner product Ó , Ô on V* by Óå, ∫Ô = ÓA, BÔ where A, B ∞ V are the vectors corresponding to the 1-forms å, ∫ ∞ V*. Let us write an arbitrary basis vector eá in terms of its components relative to the basis {eá} as eá = ∂jáeé. Therefore, in the above isomorphism, we may define the linear form eÄá ∞ V* corresponding to the basis vector eá by eÄá = géÉ ∂jáøk = g ikøk and hence using the inverse matrix, we find that ø k = g kieÄá . Applying our definition of the inner product in V* we have ÓeÄá, eÄéÔ = Óeá, eéÔ = gáé, and therefore we obtain Ó!i,!!jÔ=Ógirˆer,!gjsˆesÔ=girgjsÓˆer,!ˆesÔ=girgjsgrs=gir"jr=gij which is the analogue in V* of the definition gáé = Óeá, eéÔ in V. Lastly, since øù j = (aî) jáøi, we see that gij=Ó!i,!!jÔ=Ó(a"1)ir!r,!(a"1)js!sÔ=(a"1)ir(a"1)jsÓ!r,!!sÔ=(a"1)ir(a"1)jsgrs so the scalars g ij may be considered to be the components of a symmetric tensor G ∞ T2 º(V) defined as above by G = g ijeá · eé. Now let g = Ó , Ô be an arbitrary (i.e., possibly degenerate) real symmetric bilinear form on the inner product space V. It follows from the cor ollary to Theorem 9.14 that there exists a basis {eá} for V in which the matrix (gáé) of g takes the unique diagonal form 11.10 THE METRIC TENSOR AN D VOLUME FORMS 613 gij=!Ir!Is0t"#$$$%&''' where r + s + t = dim V = n. Thus g(ei,!ei)=!!1for 1!i!r"1for r+1!i!r+s!!0for r+s+1!i!n#$%&%!!. If r + s < n, the inner product is degene rate and we say that the space V is singular (with respect to the given inner product). If r + s = n, then the inner product is nondegenerate, and the basis {eá} is orthonormal. In the orthonor - mal case, if either r = 0 or r = n, the space is said to be ordinary Euclidean , and if 0 < r < n, then the space is called pseudo -Euclidean . Recall that the number r - s = r - (n - r) = 2r - n is called the signature of g (which is therefore just the trace of (gáé)). Moreover, the number of -1’s is called the index of g, and is denoted by Ind(g). If g = Ó , Ô is to be a metric on V, then by definition, we must have r + s = n so that t he inner product is nondegenerate. In this case, the basis {eá} is called g-orthonormal . Example 11.13 If the metric g represents a positive definite inner product on V, then we must have Ind(g) = 0, and such a metric is sai d to be Riemannian . Alternatively, another well-known metric is the Lorentz metric used in the theory of special relativity. By definition, a Lorentz metric η has Ind( η) = 1. Therefore, if η is a Lorentz metric, an η-orthonormal basis {eè, . . . , eñ} ordered in such a way that η(eá, eá) = +1 for i = 1, . . . , n - 1 and η(eñ, eñ) = -1 is called a Lorentz frame . Thus, in terms of a g-orthonormal basis, a Riemannian metric has the form (gij)=10!001!0"""00!1!"###$%&&& while in a Lorentz frame, a Lorentz metric takes the form (!ij)=10!!!001!!!0""!!"00!"1#$%%%&'(((!!. MULTILINEAR MAPPINGS AND TENSORS 614 It is worth remarking that a Lorentz metric is also frequently defined as having Ind( η) = n - 1. In this case we have η(eè, eè) = 1 and η(eá, eá) = -1 for each i = 2, . . . , n. We also point out that a vector v ∞ V is called timelike if η(v, v) < 0, lightlike (or null) if η(v, v) = 0, and spacelike if η(v, v) > 0. Note that a Lorentz inner product is clearly indefinite since, for example, the nonzero vector v with components v = (0, 0, 1, 1) has the property that Óv, vÔ = η(v, v) = 0. ∆ We now show that introducin g a metric on V leads to a unique volume form on V. Theorem 11.20 Let g be a metric on an n-dimensional oriented vector space (V, [ø]). Then, corresponding to the metric g, there exists a unique volume form µ = µ(g) ∞ [ø] such that µ(eè , . . . , eñ) = 1 for every positively oriented g-orthonormal basis {eá} for V. Moreover, if {vá} is any (not necessarily g- orthonormal ) positively oriented basis for V with dual basis {v i}, then µ = \det(g(vá, vé)) \1/2 v1° ~ ~ ~ °vn . In particular, if {vá} = {eá } is a g-orthonormal basis, then µ = e 1° ~ ~ ~ °en. Proof Since ø ≠ 0, there exists a positively oriented g-orthonormal basis {eá} such that ø(eè, . . . , eñ) > 0 (we can multiply eè by -1 if necessary in order that {eá} be positively oriented). We now define µ ∞ [ø] by µ(eè, . . . , eñ) = 1 . That thi s defines a unique µ follows by multilinearity. We claim that if {fá} is any other positively oriented g-orthonormal basis, then µ(fè, . . . , fñ) = 1 also. To show this, we first prove a simple general result. Suppose {vá} is any other basis for V relate d to the g-orthonormal basis {eá} by vá = ƒ(eá) = eéa já where, by Theorem 11.17, we have det ƒ = det(a ié). We then have g(vá, vé ) = a ráaség(e r, es) which in matrix notation is [g] v = AT[g]eA, and hence det(g(vi,!vj))=(det!)2det(g(er,!es))!!. (9) However, since {e á} is g-orthonormal we have g(e r, es) = ±∂rs, and therefore \det(g(e r, es))\ = 1. In other words |det(g(vi,!vj))|1/2=|det!|!!. (10) 11.10 THE METRIC TENSOR AN D VOLUME FORMS 615 Returning to our problem, we have det(g(fá, fé)) = ±1 also since {fá} = {ƒ(eá)} is g-orthonormal . Thus (10) implies that \det ƒ \ = 1. But {fá} is posi- tively oriented so that µ(fè, . . . , fñ) > 0 by definition. Therefore 0<µ(f1,!…!,!fn)=µ(!(e1),!…!µ(!(en))=(!*µ)(e1,!…!,en)=(det!)µ(e1,!…!,!en)=det! so that we must in fact have det ƒ = +1. In other words, µ(fè, . . . , fñ) = 1 as claimed. Now suppose that {vá} is an arbitr ary positively oriented basis for V such that vá = ƒ(eá). Then, analogously to what we have just shown, we see that µ(vè, . . . , vñ) = det ƒ > 0. Hence (10) shows that (using Example 11.8) µ(v1,!…!,!vn)=det!=\det(g(vi,!vj))|1/2=|det(g(vi,!vj))|1/2v1"!"vn(v1,!…!,!vn) which implies µ = \det(g(vá, vé))\1/2 v1° ~ ~ ~ °vn . ˙ The unique volume form µ defined in Theorem 11.20 is called the g- volume , or sometimes the metric volume form . A common (although rather careless) notation is to write \det(g(vá, vé))\1/2 = “ \”g”\” . In this notation, the g- volume is written as “ \”g”\” v1° ~ ~ ~ °vn where {vè, . . . , vñ} must be positively oriented. If the basis {vè, vì, . . . , vñ} is negatively oriented, then clearly {vì, vè, . . . , vñ} will be positively oriented. Furthermore, even though the matrix of g rela tive to each of these oriented bases will be different, the deter minant actually remains unchanged (see the discussion following the corollary to Theorem 11.13). Therefore, for this negatively oriented basis, the g-volume is “ \”g”\” v2°v1°~ ~ ~°v n = -“ \”g”\” v1°v2° ~ ~ ~ °vn . We thus have the following corollary to Theorem 11.20. Corollary Let {vá} be any basis for the n-dimensional oriented vector space (V, [ø]) with metric g. Then the g-volume form on V is given by ±“ \”g”\” v1°~ ~ ~°v n MULTILINEAR MAPPINGS AND TENSORS 616 where the “+” sign is for {vá} positively oriented, and the “-” sign is for {vá} negatively oriented. Example 11.14 From Example 11.13, we see that for a Riemannian metric g and g-orthonormal basis {eá } we have det(g(eá, eé)) = +1. Hence, from equa - tion (9), we see that det(g(vá, vé)) > 0 for any basis {vá = ƒ(eá)}. Thus the g- volume form on a Riemannian space is given by ±“g” v1° ~ ~ ~ °vn. For a Lorentz metric we have det(¨(eá, eé)) = -1 in a Lorentz frame, and therefore det(g(vá, vé)) < 0 in an arbitrary frame. Thus the g-volume in a Lorentz space is given by ±“ -”g” v1° ~ ~ ~ °vn. Let us point out that had we defined Ind( η) = n - 1 instead of Ind( η) = 1, then det( η(eá, eé)) < 0 only in an even dimensional space. In this case, we would have to write the g-volume as in the above corollary. ∆ Example 11.15 (This example is a continuation of Example 11.11.) We remark that these volume elements are of great practical use in the theory of integration on manifolds. To see an example of how this is done, let us use Examples 11.1 and 11.11 to write the volume element as (remember that this applies only locally, and hence the met ric depends on the coordinates) d† = “ \”g”\” dx1° ~ ~ ~ °dx n . If we go to a new coordinate system {xõ i}, then gij=!xr!xi!xs!xjgrs so that \gõ\ = (Jî) 2\g\ where Jî = det($x r/$xõi) is the determinant of the inverse Jacobian matrix of the transformation. But using dxõ i = ($xõ i/$xj)dxj and the properties of the wedge product, it is easy to see that dx1!!!dxn="x1"xi1!!!"xn"xindxi1!!!dxin=det"xi"xj#$%%&'((dx1!!!dxn and hence dxõ1°~ ~ ~°dxõ n = J dx1° ~ ~ ~ °dx n 11.10 THE METRIC TENSOR AN D VOLUME FORMS 617 where J is the determinant of the Jacobian matrix. (Note that the proper transformation formula for the volume element in multiple integrals arises naturally in the algebra of exterior forms.) We now have d!=|g|dx1"!"dxn=J#1|g|Jdx1"!"dxn=|g|dx1"!"dxn=d! and hence d† is a scalar called the invariant volume element . In the case of ®4 as a Lorentz space, this result is used in the theory of relativity. ∆ Exercises 1. Suppose V has a metric gáé defined on it. Show that for any A, B ∞ V we have ÓA, BÔ = aáb i = a ibá. 2. According to the special theory of r elativity, the speed of light is the same for all unaccelerated observers regardless of the motion of the source of light relative to the observer. Consider two observers moving at a constant velocity ∫ with respect to each other, and assume that the origi ns of their respective coordinate systems coincide at t = 0. If a spherical pulse of light is emitted from the origin at t = 0, then (in units where the speed of light is equal to 1) this pulse satisfies the equation x 2 + y 2 + z 2 - t2 = 0 for the first observer, and xõ 2 + yõ 2 + zõ 2 - t ä2 = 0 for the second observer. We shall use the common notation (t, x, y, z) = (x 0, x1, x2, x3) for our coordinates, and hence the Lorentz metric takes the form !µ"=#1111$%&&&'())) where 0 ¯ µ, ¥ ¯ 3. (a) Let th e Lorentz transformation matrix be Ò so that xõ µ = Ò µ¥ x¥. Show that the Lorentz transformation must satisfy Ò T ηÒ = η. (b) If the {xõ µ} system moves along the x 1-axis with velocity ∫, then it turns out that the Lorentz transformation is given by xõ0 = ©(x 0 - ∫x1) xõ1 = ©(x 1 - ∫x0) MULTILINEAR MAPPINGS AND TENSORS 618 xõ2 = x 2 xõ3 = x 3 where © 2 = 1/(1 - ∫2). Using Ò µ¥ = $xõ µ/$x¥, write out the matrix (Ò µ¥), and verify explicitly that Ò TηÒ = η. (c) The electromagnetic field tensor is given by Fµ!=0"Ex"Ey"EzEx!0!Bz"ByEy"Bz!0!BxEz!By"Bx!0#$%%%%&'((((!!. Using this, find the components of the electric field E ë and magnetic field Bë in the {xõ µ} coordinate system. In other words, find Fä µ¥ . (The actual definition of F µ¥ is given by F µ¥ = $ µA¥ - $¥Aµ where $ µ = $/$x µ and Aµ = (ƒ, A 1, A2, A3) is related to Eë and Bë through the classical equations Eë = -#ƒ - $Aë/$t and Bë = # ª Aë. See also Exercise 11.1.6.) 3. Let V be an n-dimensional vector space with a Lorentz metric η, and let W be an (n - 1)-dimensional subspace of V. Note that WÊ = {v ∞ V: η(v, w) = 0 for all w ∞ W} is the 1-dimensional subspace of all normal vectors for W. We say that W is timelike if every normal vector is spacelike , null if every normal vector is null, and spacelike if every normal vector is timelike. Prove that η restricted to W is (a) Positive definite if W is spacelike. (b) A Lorentz metric if W is timelike. (c) Degenerate if W is null. 4. (a) Let D be a 3 x 3 determinant considered as a function of three contra - variant vectors A i(1), Ai(2), and A i(3). Show that under a change of coordi - nates , D does not transform as a scalar, but that D“ \”g”\” does tra nsform as a proper scalar. [ Hint: Use Exercise 11.2.8.] (b) Show that ´ ijk“ \”g”\” transforms like a tensor. (This is the Levi-Civita tensor in general coordinates. Note that in a g-orthonormal coordinate system this reduces to the Levi-Civita symbol.) (c) What is the contravariant version of the tensor in part (b)?