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. 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. will be \(\nabla_{X} T = … 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. Rank 1 Tensors (Vectors) The definitions for contravariant and covariant tensors are inevitably defined at the beginning of all discussion on tensors. This correction term is easy to find if we consider what the result ought to be when differentiating the metric itself. 4 0 obj << /S /GoTo /D [6 0 R /Fit] >> tive for arbitrary manifolds. 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. This in effect requires running Table with an arbitrary number of indices, and then adding one. 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 . A visualization of a rank 3 tensor from [3] is shown in gure 1 below. G is a second-rank contravariant tensor. Tensor transformations. endobj For example, the covariant derivative of the stress-energy tensor T (assuming such a thing could have some physical significance in one dimension!) A Riemannian space is a manifold characterized by the existing of a symmetric rank-2 tensor called the metric tensor. ... (p, q) is of type (p, q+1), i.e. 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. In that spirit we begin our discussion of rank 1 tensors. 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: The velocity vector in equation (3) corresponds to neither the covariant nor contravari- 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. %���� /Length 2333 • In N-dimensional space a tensor of rank n has Nn components. endobj 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 rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. 12 0 obj In generic terms, the rank of a tensor signi es the complexity of its structure. 1 to third or higher-order tensors is straightforward given g (see supplemental Sec. Applying this to G gives zero. The covariant derivative increases the rank of the tensor because it contains information about derivatives in all possible spacetime directions. /Filter /FlateDecode The covariant derivative of a second rank tensor … We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. Tensors In this lecture we define tensors on a manifold, and the associated bundles, and operations on tensors. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. We want to add a correction term onto the derivative operator \(d/ dX\), forming a new derivative operator \(∇_X\) that gives the right answer. 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. metric tensor, which would deteriorate the accuracy of the covariant deriva-tive and prevent its application to complex surfaces. 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. QM�*�Jܴ2٘���1M"�^�ü\�M��CY�X�MYyXV�h� The commutator of two covariant derivatives, … 3.2. We end up with the definition of the Riemann tensor and the description of its properties. For example, dx 0 can be written as . 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. 4.4 Relations between Cartesian and general tensor fields. ARTHUR S. LODGE, in Body Tensor Fields in Continuum Mechanics, 1974. Just a quick little derivation of the covariant derivative of a tensor. 3.1. In particular, is a vector field along the curve itself. 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. >> Then we define what is connection, parallel transport and covariant differential. 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. stream and similarly for the dx 1, dx 2, and dx 3. << /S /GoTo /D (section*.1) >> \(∇_X\) is called the covariant derivative. Rank-0 tensors are called scalars while rank-1 tensors are called vectors. 1 0 obj Having defined vectors and one-forms we can now define tensors. g 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. 5 0 obj 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 What the result ought to be when differentiating the metric tensor, which rank! 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