derivative for an arbitrary-rank tensor. %PDF-1.4 %���� tive for arbitrary manifolds. 5 0 obj H�b```f`�(c`g`��� Ā B@1v�>���� �g�3U8�RP��w(X�u�F�R�D�Iza�\*:d$,*./tl���u�h��l�CW�&H*�L4������'���,{z��7҄�l�C���3u�����J4��Kk�1?_7Ϻ��O����U[�VG�i�qfe�\0�h��TE�T6>9������(V���ˋ�%_Oo�Sp,�YQ�Ī��*:{ڛ���IO��:�p�lZx�K�'�qq�����/�R:�1%Oh�T!��ۚ���b-�V���u�(��%f5��&(\:ܡ�� ��W��òs�m�����j������mk��#�SR. Tensors In this lecture we deﬁne tensors on a manifold, and the associated bundles, and operations on tensors. Notice that the Lie derivative is type preserving, that is, the Lie derivative of a type (r,s) tensor is another type (r,s) tensor. From one covariant set and one con-travariant set we can always form an invariant X i AiB i = invariant, (1.12) which is a tensor of rank zero. it has one extra covariant rank. 12 0 obj It follows at once that scalars are tensors of rank (0,0), vectors are tensors of rank (1,0) and one-forms are tensors of rank (0,1). A tensor of rank (m,n), also called a (m,n) tensor, is deﬁned to be a scalar function of mone-forms and nvectors that is linear in all of its arguments. symbol which involves the derivative of the metric tensors with respect to spacetime co-ordinate xµ(1,x2 3 4), Γρ αβ = 1 2 gργ(∂gγα ∂xβ + ∂gγβ α − ∂gαβ ∂xγ), (6) which is symmetric with respect to its lower indices. The covariant derivative increases the rank of the tensor because it contains information about derivatives in all possible spacetime directions. We also provide closed-form expressions of pairing, inner product, and trace for this discrete representation of tensor ﬁelds, and formulate a discrete covariant derivative We end up with the definition of the Riemann tensor and the description of its properties. In that spirit we begin our discussion of rank 1 tensors. In most standard texts it is assumed that you work with tensors expressed in a single basis, so they do not need to specify which basis determines the densities, but in xAct we don't assume that, so you need to be specific. g Rank-0 tensors are called scalars while rank-1 tensors are called vectors. ... (p, q) is of type (p, q+1), i.e. /Length 2333 1 The index notation Before we start with the main topic of this booklet, tensors, we will ﬁrst introduce a new notation for vectors and matrices, and their algebraic manipulations: the index To define a tensor derivative we shall introduce a quantity called an affine connection and use it to define covariant differentiation. As far as I can tell, the covariant derivative of a general higher rank tensor is simply deﬁned so that it contains terms as speciﬁed here. The covariant derivative rw of a 1-form w returns a rank-2 tensor whose symmetric part is the Killing operator of w, i.e., 1 2 rw+rwt..=K(w).yThe Killing operator is, itself, remarkably relevant in differential geometry: its kernel corresponds to vector ﬁelds (known as The expression in the case of a general tensor is: (3.21) It follows directly from the transformation laws that the sum of two connections is not a connection or a tensor. will be \(\nabla_{X} T = … << %PDF-1.5 Definition: the rank (contravariant or covariant) of a tensor is equal to the number of components: Tk mn rp is a mixed tensor with contravariant rank = 4 and covariant rank = 2. The covariant derivative of a second rank tensor … A visualization of a rank 3 tensor from [3] is shown in gure 1 below. Examples 1. Higher-order tensors are multi-dimensional arrays. Then we define what is connection, parallel transport and covariant differential. Having deﬁned vectors and one-forms we can now deﬁne tensors. 50 0 obj << /Linearized 1 /O 53 /H [ 2166 1037 ] /L 348600 /E 226157 /N 9 /T 347482 >> endobj xref 50 79 0000000016 00000 n 0000001928 00000 n 0000002019 00000 n 0000003203 00000 n 0000003416 00000 n 0000003639 00000 n 0000004266 00000 n 0000004499 00000 n 0000005039 00000 n 0000025849 00000 n 0000027064 00000 n 0000027620 00000 n 0000028837 00000 n 0000029199 00000 n 0000050367 00000 n 0000051583 00000 n 0000052158 00000 n 0000052382 00000 n 0000053006 00000 n 0000068802 00000 n 0000070018 00000 n 0000070530 00000 n 0000070761 00000 n 0000071180 00000 n 0000086554 00000 n 0000086784 00000 n 0000086805 00000 n 0000088020 00000 n 0000088115 00000 n 0000108743 00000 n 0000108944 00000 n 0000110157 00000 n 0000110453 00000 n 0000125807 00000 n 0000126319 00000 n 0000126541 00000 n 0000126955 00000 n 0000144264 00000 n 0000144476 00000 n 0000145196 00000 n 0000145800 00000 n 0000146420 00000 n 0000147180 00000 n 0000147201 00000 n 0000147865 00000 n 0000147886 00000 n 0000148542 00000 n 0000166171 00000 n 0000166461 00000 n 0000166960 00000 n 0000167171 00000 n 0000167827 00000 n 0000167849 00000 n 0000179256 00000 n 0000180483 00000 n 0000181399 00000 n 0000181602 00000 n 0000182063 00000 n 0000182750 00000 n 0000182772 00000 n 0000204348 00000 n 0000204581 00000 n 0000204734 00000 n 0000205189 00000 n 0000206409 00000 n 0000206634 00000 n 0000206758 00000 n 0000222032 00000 n 0000222443 00000 n 0000223661 00000 n 0000224303 00000 n 0000224325 00000 n 0000224909 00000 n 0000224931 00000 n 0000225441 00000 n 0000225463 00000 n 0000225542 00000 n 0000002166 00000 n 0000003181 00000 n trailer << /Size 129 /Info 48 0 R /Root 51 0 R /Prev 347472 /ID[<5ee016cf0cc59382eaa33757a351a0b1>] >> startxref 0 %%EOF 51 0 obj << /Type /Catalog /Pages 47 0 R /Metadata 49 0 R /AcroForm 52 0 R >> endobj 52 0 obj << /Fields [ ] /DR << /Font << /ZaDb 44 0 R /Helv 45 0 R >> /Encoding << /PDFDocEncoding 46 0 R >> >> /DA (/Helv 0 Tf 0 g ) >> endobj 127 0 obj << /S 820 /V 1031 /Filter /FlateDecode /Length 128 0 R >> stream %���� To find the correct transformation rule for the gradient (and for covariant tensors in general), note that if the system of functions F i is invertible ... Now we can evaluate the total derivatives of the original coordinates in terms of the new coordinates. The rank of a tensor is the total number of covariant and contravariant components. In particular, is a vector field along the curve itself. In generic terms, the rank of a tensor signi es the complexity of its structure. In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In later Sections we meet tensors of higher rank. In xTensor you need to tell the system in advance that the derivative will add density terms for tensor densities in a given basis. If we apply the same correction to the derivatives of other second-rank contravariant tensors, we will get nonzero results, and they will be the right nonzero results. The relationship between this and parallel transport around a loop should be evident; the covariant derivative of a tensor in a certain direction measures how much the tensor changes relative to what it would have been if it had been parallel transported (since the covariant derivative of a tensor in a direction along which it is parallel transported is zero). Tensors of rank 0 are scalars, tensors of rank 1 are vectors, and tensors of rank 2 are matrices. I'm keeping track of which indices are contravariant/upper and covariant/lower, so the problem isn't managing what each term would be, but rather I'm having difficulty seeing how to take an arbitrary tensor and "add" a new index to it. 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