Applying this to G gives zero. Generalizing the norm structure in Eq. 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. << /S /GoTo /D (section*.1) >> G is a second-rank contravariant tensor. In particular, is a vector field along the curve itself. To define a tensor derivative we shall introduce a quantity called an affine connection and use it to define covariant differentiation. rotations between Cartesian systems: Sj k = ½ [ δrj/ xk - δrk/ … %PDF-1.5 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. endobj endobj 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 /Filter /FlateDecode 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: 3.1. 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. In general, if a tensor appears to vary, it could vary either because it really does vary or because the … 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. 12 0 obj 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. 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 rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. This correction term is easy to find if we consider what the result ought to be when differentiating the metric itself. (\376\377\000P\000i\000n\000g\000b\000a\000c\000k\000s) We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. 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. tive for arbitrary manifolds. g 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 A visualization of a rank 3 tensor from [3] is shown in gure 1 below. In that spirit we begin our discussion of rank 1 tensors. 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. 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 derivative for an arbitrary-rank tensor. 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. For example, dx 0 can be written as . 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). Examples 1. In xTensor you need to tell the system in advance that the derivative will add density terms for tensor densities in a given basis. The velocity vector in equation (3) corresponds to neither the covariant nor contravari- 5 0 obj The rank of a tensor is the total number of covariant and contravariant components. 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. 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. 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. 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. where the covariant quantities transform cogrediently to the basis vectors and the con-travariant quantities transform contragrediently. 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}. Given the … /Length 2333 Having deﬁned vectors and one-forms we can now deﬁne tensors. 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 . Remark 2.2. Tensor transformations. 50 0 obj
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\(∇_X\) is called the covariant derivative. 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 QM�*�Jܴ2٘���1M"�^�ü\�M��CY�X�MYyXV�h� The covariant derivative increases the rank of the tensor because it contains information about derivatives in all possible spacetime directions. We want to add a correction term onto the derivative operator \(d/ dX\), forming a new derivative operator \(∇_X\) that gives the right answer. We end up with the definition of the Riemann tensor and the description of its properties. 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). it has one extra covariant rank. metric tensor, which would deteriorate the accuracy of the covariant deriva-tive and prevent its application to complex surfaces. 3.2. endobj %���� %PDF-1.4
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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. 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. For example, the metric tensor, which has rank two, is a matrix. >> 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. Rank 1 Tensors (Vectors) The deﬁnitions for contravariant and covariant tensors are inevitably deﬁned at the beginning of all discussion on tensors. As an application, the Helmholtz-Hodge decomposition (HHD) is adapted in the context of the Galerkin method for displaying the irrotational, incompressible, and har-monic components of vectors on curved surfaces. Then we define what is connection, parallel transport and covariant differential. 4 0 obj ARTHUR S. LODGE, in Body Tensor Fields in Continuum Mechanics, 1974. 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. 1 to third or higher-order tensors is straightforward given g (see supplemental Sec. The commutator of two covariant derivatives, … 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. << 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. 1 0 obj �E]x�������Is�b�����z��٪�
2yc��2�����:Z��.,�*JKM��M�8� �F9�95&�ڡ�.�3����. 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 of rank 0 are scalars, tensors of rank 1 are vectors, and tensors of rank 2 are matrices. • In N-dimensional space a tensor of rank n has Nn components. will be \(\nabla_{X} T = … ... (p, q) is of type (p, q+1), i.e. << /S /GoTo /D [6 0 R /Fit] >> and similarly for the dx 1, dx 2, and dx 3. 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. Just a quick little derivation of the covariant derivative of a tensor. 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. Tensors In this lecture we deﬁne tensors on a manifold, and the associated bundles, and operations on tensors. Strain tensor w.r.t. Their deﬁnitions are inviably without explanation. Rank-0 tensors are called scalars while rank-1 tensors are called vectors. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. 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. 4.4 Relations between Cartesian and general tensor fields. In later Sections we meet tensors of higher rank. The covariant derivative of a second rank tensor … Similarly, the derivative of a second order tensor h is a third order tensor ∇gh. This in effect requires running Table with an arbitrary number of indices, and then adding one. Higher-order tensors are multi-dimensional arrays. 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