Introduction Tensors over a vector space Remark The dual

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2 SVANTE JANSON, In other words the matrix hei ej i ni j 1 is the identity matrix. Note that duality of bases is a symmetric relation using the identification. of V and V the dual basis of e1 en is e1 P en, If f1 fn is another basis in V then fi j aij ej for some numbers. aij and similarly for the corresponding dual bases f i j bij ej We have. hf i fj i k bik hek fj i k bik ajk and thus introducing the matrices A. aij ij and B bij ij,I hf i fj i ij BAt, Consequently the matrices of the changes of bases are related by B At 1. 1 2 Tensors For two non negative integers k and l T k l V is defined to be. the vector space of all multilinear maps f v1 vk v1 vl V. V V V R with k arguments in V and l arguments in V The. elements of T k l V are called tensors of degree or order k l. Given a basis e1 en of V a tensor T of degree k l is uniquely deter. mined by the nk l numbers which we call coefficients of the tensor. T ei1 eik ej1 ejl 1 1, Conversely there is such a tensor of degree k l for any choice of these num. bers Thus the vector space T k l V has dimension nk l. A tensor of type k 0 for some k 1 is called contravariant and a tensor. of type 0 l for some l 1 is called covariant Similarly in the coefficients. 1 1 the indices i1 ik are called contravariant and the indices j1 jl. are called covariant It is conventional and convenient to use superscripts for. contravariant indices and subscripts for covariant indices. Example 1 2 1 The case k l 0 is rather trivial but useful T 0 0 V is. just the vector space of all real numbers T 0 0 V R and a tensor of degree. 0 0 is a real number Tensors of degree 0 0 are also called scalars. Example 1 2 2 T 0 1 V is the space of all linear maps V R Thus. T 0 1 V V and a tensor of degree 0 1 is a linear functional on V. Example 1 2 3 Similarly T 1 0 V equals V which we identify with V. Thus T 1 0 V V and a tensor of degree 1 0 is an element of V. Example 1 2 4 A linear mapping S V V defines a tensor T of degree. T v v hv S v i v V v V 1 2,In a basis e1 en this tensor has the coefficients.
Tji T ei ej hei S ej i sij, where sij is the matrix representation of S in the basis e1 en Hence. 1 2 defines a natural bijection between T 1 1 V and the space hom V V of. all linear mappings V V so we may identify T 1 1 V and hom V V and. regard a tensor of degree 1 1 as a linear mapping V V Note that the. identity mapping V V thus corresponds to the special tensor T 1 1 V. TENSORS AND DIFFERENTIAL FORMS 3, given by v v hv vi with coefficients ij ij in this context better. written ji, Example 1 2 5 More generally a multilinear mapping S V l V defines. a tensor T of degree 1 l by T v v1 vl hv S v1 vl i Hence we. may identify T 1 l V and the space hom V V V of all such multilinear. 1 3 Tensor product Tensors may be multiplied by real numbers and two. tensors of the same degree may be added because each T k l V is a vector. space Moreover there is a multiplication known as tensor product such that. any two tensors may be multiplied The tensor product of two tensors T and U. of degrees k1 l1 and k2 l2 respectively is a tensor of degree k1 k2 l1 l2. denoted by T U and defined by,T U v1 vk 1 k2 v1 vl1 l2. T v1 vk 1 v1 vl1 U vk 1 1 vk 1 k2 vl1 1 vl1 l2 1 3. If T or U has order 0 0 the tensor product reduces to the ordinary multi. plication of a tensor by a real number, By 1 1 and 1 3 the coefficients of the tensor product are given by.
v1 v v1 v v v,T U v1 vlk11 l,Tv1 vlk11 Uvlk11 1, The tensor product is associative T1 T2 T3 T1 T2 T3 for. any three tensors T1 T2 T3 so we may write T1 T2 T3 etc without any. danger for the tensor product of three or more tensors The tensor product. is not commutative however so it is important to keep track of the order. of the tensors The tensor product is further distributive T1 T2 T3. T1 T3 T1 T3 and T1 T2 T3 T1 T2 T1 T3, Remark The tensor algebra is defined as the direct sum k l 0 T V of. all the spaces T k l V its elements are thus sums of finitely many tensors of. different degrees Any two elements in the tensor algebra may be added or. multiplied so it is an algebra in the algebraic sense. We will not use this construct which sometimes is convenient for our pur. poses it is better to consider the spaces T k l V separately. Remark There is also a general construction of the tensor product of two. vector spaces We will not define it here but remark that using it T k l V. V V V V with k factors V and l factors V which yields. another and perhaps more natural definition of the tensor spaces T k l V. 1 4 Bases If e1 en is a basis in V we can for any indices i1 ik. j1 jl 1 n for the tensor product,ei1 eik ej1 ejl T k l V 1 4. Recall that ei T 1 0 V and ej T 0 1 V It follows from 1 1 and 1 3. that this tensor has coefficients, ei1 eik ej1 ejl ei1 eik ej10 ejl0 i1 i01 i2 i02 jl jl0. 4 SVANTE JANSON, in the basis e1 en thus exactly one coefficient is 1 and the others 0 It.
follows that the set of the nk l tensors ei1 eik ej1 ejl form a. basis in T k l V and that the coefficients 1 1 of a tensor T T k l V are the. coordinates in this basis i e,e eik ej1 ejl,i1 ik j1 jl. 1 5 Change of basis Consider again a change of basis fi j aij ej and. thus for the dual bases f i j,j bij e where as shown above the matrices. A aij ij and B bij ij are related by B At 1 A 1 t It follows. immediately from 1 1 that the coefficients of a tensor T in the new basis are. T f i1 f ik fj1 fjl T bi1 p1 ep1 ajl ql eql,bi1 p1 bik pk aj1 q1 ajl ql Tqp11 q. p1 pk q1 ql, Thus all covariant indices are transformed using the matrix A aij ij and. the contravariant indices using B bij ij, Remark This is the historical origin of the names covariant and contravariant.
The names have stuck although in the modern point of view with empha. sis on vectors and vector spaces rather than coefficients they are really not. appropriate, Remark It may be better to write ai j and bi j to adhere to the convention that. a summation index usually appears once as a subscript and once as a super. script and that other indices should appear in the same place on both sides of. an equation In fact many authors use the Einstein summation convention. where summation signs generally are omitted and a summation is implied for. each index repeated this way, Example 1 5 1 Both linear operators V V and bilinear forms on V may be. represented by matrices but as is well known from elementary linear algebra. the matrices transform differently under changes of basis We see here the. reason a linear operator is a tensor of degree 1 1 but a bilinear form is a. tensor of degree 0 2, 1 6 Contraction A tensor T of degree 1 1 may be regarded as a linear. operator in V Example 1 2 4 The trace of this linear operator is called the. contraction of T, More generally let T be a tensor of degree k l with k l 1 and select. one contravariant and one covariant argument By keeping all other arguments. fixed T becomes a bilinear map V V R of the two selected arguments i e. a tensor of degree 1 1 which has a contraction i e trace This contraction. is a real number depending on the remaining k 1 l 1 arguments and. TENSORS AND DIFFERENTIAL FORMS 5, it is clearly multilinear in them so it defines a tensor of degree k 1 l 1.
again called the contraction of T, Given a basis e1 en and its dual basis e1 en the contraction of. a tensor T of degree 1 1 is given by letting S denote the corresponding linear. operator as in 1 2,hei S ei i T ei ei Tii, More generally the coefficients of a contraction of a tensor T of degree k l are. obtained by summing the n coefficients of T where the indices corresponding. to the two selected arguments are equal For example contracting the second. contravariant and first covariant indices or arguments in a tensor T of degree. 2 3 we obtain a tensor with coefficients Tejk m Tmjk. Note that we have to specify the indices or arguments that we contract A. tensor of degree k l has kl contractions all of the same degree k 1 l 1. but in general different, 1 7 Symmetric and antisymmetric tensors A covariant tensor T of de. gree 2 is symmetric if T v w T w v for all v w V and antisymmetric. if T v w T w v for all v w V The terms alternating and skew. are also used for the latter More generally a covariant tensor T of de. gree k is symmetric if T v 1 v k T v1 vk and antisymmetric if. T v 1 v k sgn T v1 vk for all v1 vk V and all permu. tations Sk the set of all k permutations of 1 k where sgn is 1. is is an even permutation and 1 is is odd Equivalently T is symmetric. antisymmetric if T v1 vk is unchanged changes its sign whenever two. of the arguments are interchanged In particular if T is antisymmetric then. T v1 vk 0 when two of v1 vk are equal In fact this property is. equivalent to T being antisymmetric, Given a basis of V a tensor T is symmetric antisymmetric if and only if. its coefficients Ti1 ik are symmetric antisymmetric in the indices Con. sequently a symmetric covariant tensor of degree k is determined by the. coefficients Ti1 ik with 1 i1 ik n and an antisymmetric. covariant tensor of degree k is determined by the coefficients Ti1 ik with. 1 i1 ik n since coefficients with two equal indices automati. cally vanish These coefficients may be chosen arbitrarily and it follows that. the linear, space of all symmetric covariant tensors of degree k has dimension.
while the linear space of all antisymmetric covariant tensors of degree. k has dimension k, Symmetric and antisymmetric contravariant tensors are defined in the same. way More generally we may say that a possibly mixed tensor is symmetric. or antisymmetric in two or several specified covariant indices or in two. or several specified contravariant indices It does not make sense to mix. covariant and contravariant indices here, The tensor product of two symmetric or antisymmetric tensors is in general. neither symmetric nor antisymmetric It is however possible to symmetrize. 6 SVANTE JANSON, antisymmetrize it and thus define a symmetric tensor product and an anti. symmetric tensor product, The antisymmetric case which is the most important in differential geome. try is studied in some detail in the next subsection. 1 8 Exterior product Let Ak V be the linear space of all antisymmetric. covariant tensors of degree k By the last subsection dim Ak V nk in. particular the case k n is trivial with dim Ak V 0 and thus Ak V 0. Note the special cases A0 V R and A1 V V Note also that An V is. one dimensional L Ln, Form further the direct sum A V k 0 A k V k 0 Ak V with.
dimension n0 nk 2n, If T Ak V and U Al V with k l 0 we define their exterior product. to be the tensor T U Ak l V given by,T U v1 vk l sgn T v 1 v k U v k 1 v k l. Remark Some authors prefer the normalization factor k l. instead of k l, which leads to different constants in some formulas. It is easily seen that when k or l equals 0 the exterior product coincides. with the usual product of a tensor and a real number. The exterior product is associative T1 T2 T3 T1 T2 T3 and. bilinear i e it satisfies the distributive rules and it may be extended to a. product on A V by bilinearity which makes A V into an associative but. not commutative algebra, The exterior product is anticommutative in the sense that. U T 1 kl T U T Ak V U Al V 1 7, If v1 vk V A1 V then v1 vk Ak V and it follows from.
1 6 and induction that,v1 vk v1 vk det hvi vj i ki j 1 1 8. If e1 en is a basis in V and e1 en is the dual basis then for. any k 0 ei1 eik 1 i1 ik n is a basis in Ak V for k 0. we interpret the empty product as 1 and any T Ak V has the expansion. T Ti1 ik ei1 eik, as is easily verified using 1 8 The set of all 2n products ei1 eik where. k is allowed to range form 0 to n is thus a basis in A V. Example 1 8 1 In particular the one dimensional space An V where as. usually n dim V is spanned by e1 en for any basis e1 en in V. TENSORS AND DIFFERENTIAL FORMS 7, If f1 fn is another basis in V with fi j aij ej then the basis change. in An V is given by,f1 fn aiji ej1 ejn,ai i e 1 e n. ai i sgn e1 en,det A e1 en 1 9,using the matrix A aij ij.
1 9 Tensors over a Euclidean space An inner product on V is a bilinear. form V V R that is positive definite and thus symmetric in other words. a special tensor of degree 0 2, Now suppose that V is a Euclidean space i e a vector space equipped with. a specific inner product which we denote both by h i and g Then there is a. natural isomorphism between V and V where v V can be identified with the. linear functional h v w 7 hv wi Using this identification we define a dual. inner product g in V by g v w g v w when v h v w h w. thus making h an isometry Note that g is a bilinear form on V and thus a. contravariant tensor of degree 2 0 We use the alternative notation h i for. g too thus h i is really used in three senses viz for the inner products in. V and V and for the pairing between V and V There is no great danger. of confusion however because the three interpretations coincide if we use the. identification between V and V hv wi hh v wi hh v h w i. Given a basis e1 en we denote the coefficients of the inner products. g and g by gij and g ij respectively If v V corresponds to v h v V. then the coefficients of v and v are related by, vi hv ei i hv ei i h v j ej ei i v j g ej ei gij v j 1 10. We thus obtain the coefficients of v by multiplying t. TENSORS AND DIFFERENTIAL FORMS SVANTE JANSON UPPSALA UNIVERSITY Introduction The purpose of these notes is to give a quick course on tensors in general di erentiable manifolds as a complement to standard textbooks Most proofs are quite straightforward and are left as exercises to the reader

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