We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. x��]�۶�ݿBo��'�� \��:�9��cg�j�N�J��8�H���|��,(�:P�5nf�p�.��X~=}�7i:���$a��rDEB8�#��q:�.F���y?N��1�I�������x�Y}{us��c�D���p���^����7���gӫw��|�g� ���'���M�"�bH�oy��x���4�6w�$z��y��nM��w�����E?�Gʉ << /S /GoTo /D [6 0 R /Fit] >> 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. Since the covariant derivative of a tensor field at a point depends only on value of the vector field at one can define the covariant derivative along a smooth curve in a manifold: Note that the tensor field only needs to be defined on the curve for this definition to make sense. A Riemannian space is a manifold characterized by the existing of a symmetric rank-2 tensor called the metric tensor. endobj 1 to third or higher-order tensors is straightforward given g (see supplemental Sec. In that spirit we begin our discussion of rank 1 tensors. Just a quick little derivation of the covariant derivative of a tensor. • In N-dimensional space a tensor of rank n has Nn components. The components of this tensor, which can be in covariant (g ij) or contravariant (gij) forms, are in general continuous variable functions of coordi- nates, i.e. 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. %PDF-1.4 %���� \(∇_X\) is called the covariant derivative. %���� To define a tensor derivative we shall introduce a quantity called an affine connection and use it to define covariant differentiation. Generalizing the norm structure in Eq. Having deﬁned vectors and one-forms we can now deﬁne tensors. 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. /Length 2333 3.1 Covariant derivative In the previous chapter we have shown that the partial derivative of a non-scalar tensor is not a tensor (see (2.34)). Tensors In this lecture we deﬁne tensors on a manifold, and the associated bundles, and operations on tensors. 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. Then we define what is connection, parallel transport and covariant differential. In general, taking a derivative of a tensor increases its order by one: The derivative of function f is a vector, a first-order tensor. metric tensor, which would deteriorate the accuracy of the covariant deriva-tive and prevent its application to complex surfaces. Rank 1 Tensors (Vectors) The deﬁnitions for contravariant and covariant tensors are inevitably deﬁned at the beginning of all discussion on tensors. �E]x�������Is�b�����z��٪� 2yc��2�����:Z��.,�*JKM��M�8� �F9�95&�ڡ�.�3����. tive for arbitrary manifolds. This correction term is easy to find if we consider what the result ought to be when differentiating the metric itself. 12.1 Basic deﬁnitions We have already seen several examples of the idea we are about to introduce, namely linear (or multilinear) operators acting on vectors on M. For example, the metric is a bilinear operator which takes two vectors to give a real number, i.e. 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 G is a second-rank contravariant tensor. << If you are in spacetime and you are using coordinates [math]x^a[/math], the covariant derivative is characterized by the Christoffel symbols [math]\Gamma^a_{bc}. and similarly for the dx 1, dx 2, and dx 3. 4 0 obj 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. (\376\377\000P\000i\000n\000g\000b\000a\000c\000k\000s) continuous 2-tensors in the plane to construct a ﬁnite-dimensional encoding of tensor ﬁelds through scalar val-ues on oriented simplices of a manifold triangulation. This in effect requires running Table with an arbitrary number of indices, and then adding one. Tensor transformations. For example, the metric tensor, which has rank two, is a matrix. So, our aim is to derive the Riemann tensor by finding the commutator or, in semi-colon notation, We know that the covariant derivative of V a is given by Also, taking the covariant derivative of this expression, which is a tensor of rank 2 we get: where the covariant quantities transform cogrediently to the basis vectors and the con-travariant quantities transform contragrediently. 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. In particular, is a vector field along the curve itself. derivative for an arbitrary-rank tensor. Higher-order tensors are multi-dimensional arrays. g Rank-0 tensors are called scalars while rank-1 tensors are called vectors. For example, for a tensor of contravariant rank 2 and covariant rank 1: T0 = @x 0 @x @x @x @xˆ @x0 T ˆ where the prime symbol identi es the new coordinates and the transformed tensor. Tensors of rank 0 are scalars, tensors of rank 1 are vectors, and tensors of rank 2 are matrices. endobj 4.4 Relations between Cartesian and general tensor fields. ARTHUR S. LODGE, in Body Tensor Fields in Continuum Mechanics, 1974. 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 Remark 2.2. 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. 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. In generic terms, the rank of a tensor signi es the complexity of its structure. >> ... (p, q) is of type (p, q+1), i.e. 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). 5 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). 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. rotations between Cartesian systems: Sj k = ½ [ δrj/ xk - δrk/ … QM�*�Jܴ2٘���1M"�^�ü\�M��CY�X�MYyXV�h� Similarly, the derivative of a second order tensor h is a third order tensor ∇gh. /Filter /FlateDecode 12 0 obj 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. In the pages that follow, we shall see that a tensor may be designated as contravariant, covariant, or mixed, and that the velocity expressed in equation (2) is in its contravariant form. In general, if a tensor appears to vary, it could vary either because it really does vary or because the … A given contravariant index of a tensor can be lowered using the metric tensor g μν , and a given covariant index can be raised using the inverse metric tensor g μν . The rank of a tensor is the total number of covariant and contravariant components. See P.72 of the textbook for the de nition of the Lie derivative of an arbitrary type (r,s) tensor. endobj Given the … 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. We end up with the definition of the Riemann tensor and the description of its properties. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. 3.1 Summary: Tensor derivatives Absolute derivative of a contravariant tensor over some path D λ a ds = dλ ds +λbΓa bc dxc ds gives a tensor ﬁeld of same type (contravariant ﬁrst order) in this case. Strain tensor w.r.t. The general formula for the covariant derivative of a covariant tensor of rank one, A i, is A i, j = ∂A i /∂x j − {ij,p}A p For a covariant tensor of rank two, B ij, the formula is: B ij, k = ∂B ij /∂x k − {ik,p}B pj − {kj,p}B ip << /S /GoTo /D (section*.1) >> 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 velocity vector in equation (3) corresponds to neither the covariant nor contravari- 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. 1 0 obj For example, the covariant derivative of the stress-energy tensor T (assuming such a thing could have some physical significance in one dimension!) stream Examples 1. 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 Rank-1 tensors are called scalars while rank-1 tensors are called vectors see supplemental Sec increases the rank of manifold... 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