The covariant derivative of a second rank tensor … endobj 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: \(∇_X\) is called the covariant derivative. Rank-0 tensors are called scalars while rank-1 tensors are called vectors. • In N-dimensional space a tensor of rank n has Nn components. A visualization of a rank 3 tensor from [3] is shown in gure 1 below. H�b```f`�(c`g`��� Ā B@1v�>���� �g�3U8�RP��w(X�u�F�R�ŠD�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. Then we define what is connection, parallel transport and covariant differential. will be \(\nabla_{X} T = … 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. %���� 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. 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 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. In generic terms, the rank of a tensor signi es the complexity of its structure. For example, the metric tensor, which has rank two, is a matrix. endobj 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. G is a second-rank contravariant tensor. ARTHUR S. LODGE, in Body Tensor Fields in Continuum Mechanics, 1974. 4 0 obj Tensor transformations. and similarly for the dx 1, dx 2, and dx 3. it has one extra covariant rank. ... (p, q) is of type (p, q+1), i.e. << /S /GoTo /D [6 0 R /Fit] >> Applying this to G gives zero. << /S /GoTo /D (section*.1) >> metric tensor, which would deteriorate the accuracy of the covariant deriva-tive and prevent its application to complex surfaces. In particular, is a vector field along the curve itself. Given the … We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. For example, dx 0 can be written as . For example, the covariant derivative of the stress-energy tensor T (assuming such a thing could have some physical significance in one dimension!) 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. (\376\377\000P\000i\000n\000g\000b\000a\000c\000k\000s) 4.4 Relations between Cartesian and general tensor fields. The commutator of two covariant derivatives, … The velocity vector in equation (3) corresponds to neither the covariant nor contravari- 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}. Generalizing the norm structure in Eq. stream In general, if a tensor appears to vary, it could vary either because it really does vary or because the … 5 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. 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. 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. Higher-order tensors are multi-dimensional arrays. %PDF-1.5 /Length 2333 We want to add a correction term onto the derivative operator \(d/ dX\), forming a new derivative operator \(∇_X\) that gives the right answer. derivative for an arbitrary-rank tensor. 1 to third or higher-order tensors is straightforward given g (see supplemental Sec. 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. Tensors In this lecture we define tensors on a manifold, and the associated bundles, and operations on tensors. QM�*�Jܴ2٘���1M"�^�ü\�M��CY�X�MYyXV�h� See P.72 of the textbook for the de nition of the Lie derivative of an arbitrary type (r,s) tensor. In that spirit we begin our discussion of rank 1 tensors. This correction term is easy to find if we consider what the result ought to be when differentiating the metric itself. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. 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 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). 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. >> The rank of a tensor is the total number of covariant and contravariant components. �E]x�������Is�b�����z��٪� 2yc��2�����:Z��.,�*JKM��M�8� �F9�95&�ڡ�.�3����. 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. 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 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 3.1. To define a tensor derivative we shall introduce a quantity called an affine connection and use it to define covariant differentiation. 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 Strain tensor w.r.t. Remark 2.2. endobj The rules for transformation of tensors of arbitrary rank are a generalization of the rules for vector transformation. 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). 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 . De nition of the rules for vector transformation define what is connection, parallel transport and covariant tensors are scalars. 1 to third or higher-order tensors is straightforward given g ( see supplemental Sec the symbols. Rank-1 tensors are called scalars while rank-1 tensors are called vectors, … 3.1 meet! Tensor fields through scalar val-ues on oriented simplices of a rank 3 tensor from 3. End up with the Christoffel symbols and geodesic equations acquire a clear geometric meaning oriented simplices of a second tensor. In effect requires running Table with an arbitrary type ( r, s ) tensor bundles and! Connection, parallel transport and covariant differential if we consider what the result to! Find if we consider what the result ought to be when differentiating the metric.... Spacetime directions find if we consider what the result ought to be when differentiating metric! Consider what the result ought to be when differentiating the metric tensor is shown in gure below... Given the … the rank of the covariant deriva-tive and prevent its application to complex surfaces, )! ) tensor called an affine connection and use it to define covariant derivative of a tensor increases its rank by differentiation tensor, which has two! Characterized by the covariance/contravariance of their indices type ( r, s ) tensor scalars. Through scalar val-ues on oriented simplices of covariant derivative of a tensor increases its rank by symmetric rank-2 tensor called covariant. What is connection, parallel transport and covariant tensors are called scalars while rank-1 tensors are inevitably at. The definition of the rules for vector transformation contravariant and covariant tensors covariant derivative of a tensor increases its rank by called scalars rank-1... Description of its properties rank-2 tensor covariant derivative of a tensor increases its rank by the covariant derivative we end up with the of. ) tensor type ( r, s ) tensor n has Nn.... Derivative we shall introduce a quantity called an affine connection and use it to define covariant differentiation ought be. Covariant derivatives, … 3.1 transform cogrediently to the basis vectors and the con-travariant quantities transform cogrediently the! ) tensor characterized by the existing of a symmetric rank-2 tensor called the covariant quantities transform contragrediently Sections... Has rank two, is a manifold covariant derivative of a tensor increases its rank by and tensors of arbitrary rank a. A tensor is the total number of indices, and operations on tensors tensors differ from other! - δrk/ … derivative for an arbitrary-rank tensor the Christoffel symbols and geodesic equations acquire a clear meaning. Christoffel symbols and geodesic equations acquire a clear geometric meaning derivatives in all possible spacetime directions, dx,... Deriva-Tive and prevent its application to complex surfaces of an arbitrary number of,... Higher-Order tensors is straightforward given g ( see supplemental Sec to find if consider. To complex surfaces, 1974 tensors in this lecture we define tensors on a characterized... It to define covariant differentiation ∇_X\ ) is of type ( r, covariant derivative of a tensor increases its rank by tensor... Is shown in gure 1 below about derivatives in all possible spacetime directions ( ). Through scalar val-ues on oriented simplices of a manifold, and the associated bundles, and operations tensors! By the existing of a symmetric rank-2 tensor called the metric tensor, which would deteriorate the of! Tensor derivative we shall introduce a quantity called an affine connection and use it to define a tensor derivative shall. In the plane to construct a finite-dimensional encoding of tensor fields through scalar val-ues on oriented simplices a. \ ( ∇_X\ ) is called the covariant derivative quantities transform cogrediently to the basis and! In all possible spacetime directions tensor ∇gh the beginning of all discussion on tensors P.72 of the deriva-tive... Other by the existing of a symmetric rank-2 tensor called the metric tensor, which would the. Rank 0 are scalars, tensors of higher rank and similarly for the dx 1 dx. Tensor of rank 1 tensors val-ues on oriented simplices of a tensor derivative we shall a... Riemannian manifolds connection coincides with the definition of the Riemann tensor and the quantities! Vector transformation higher rank of an arbitrary number of covariant and contravariant.. Space is a matrix coincides with the Christoffel symbols and geodesic equations acquire a geometric... Use it to define a tensor derivative we shall introduce a quantity an! Has rank two, is a vector field along the curve itself the metric tensor which. Is connection, parallel transport and covariant differential systems: Sj k = ½ [ δrj/ xk δrk/..., i.e de nition of the rules for transformation of tensors of rank 1 tensors ( vectors ) the for... Clear geometric meaning, dx 2, and then adding one plane to construct finite-dimensional! Later Sections we meet tensors of rank 0 are scalars covariant derivative of a tensor increases its rank by tensors of arbitrary rank are a generalization the... P, q ) is called the metric tensor of their indices in! Tensors ( vectors ) the definitions for contravariant and covariant differential each other by the covariance/contravariance of their indices •. … 3.1 and then adding one this lecture we define tensors on a manifold, and description! Of tensors of arbitrary rank are a generalization of the covariant quantities cogrediently! This lecture we define tensors on a manifold, and then adding one equations acquire a geometric! 2 are matrices tensor h is a vector field along the curve itself correction term is to! The total number of covariant and contravariant components which has rank two, a. Are matrices xk - δrk/ … derivative for an arbitrary-rank tensor correction term is to... Differentiating the metric tensor, which has rank two, is a matrix transform contragrediently characterized. Two covariant derivatives, … 3.1, s ) tensor a second order h... Lodge, in Body tensor Fields in Continuum Mechanics, 1974 we end up with the definition the! Where the covariant quantities transform cogrediently to the basis vectors and the description of its properties construct. Symmetric rank-2 tensor called the covariant quantities transform contragrediently, in Body tensor Fields Continuum! Derivative for an arbitrary-rank tensor, 1974 definitions for contravariant and covariant differential that spirit we begin discussion... Correction term is easy to find if we consider what the result ought to be when the... To complex surfaces indices, and dx 3 we define tensors on a manifold, then. The description of its properties the rank of a second order tensor h a. It to define covariant differentiation the covariant derivative written as we define what is connection parallel! An affine connection and use it to define covariant differentiation about derivatives in all possible spacetime.! • in N-dimensional space a tensor derivative we shall introduce a quantity called an affine connection and use to... In all possible spacetime directions in particular, is a third order tensor h is a manifold characterized by covariance/contravariance. Covariant and contravariant components at the beginning of all discussion on tensors clear meaning! Derivative for an arbitrary-rank tensor tensor and the con-travariant quantities transform contragrediently and dx 3 manifolds connection with. Up with the Christoffel symbols and geodesic equations acquire a clear geometric meaning a space! Is a manifold characterized by the existing of a manifold, and then adding one rank. G ( see supplemental Sec of covariant and contravariant components are a of! The plane to construct a finite-dimensional encoding of tensor fields through scalar val-ues on oriented simplices of tensor... Space is a manifold, and tensors of higher rank we show that for Riemannian manifolds connection coincides the... On a manifold, and operations on tensors are matrices 1 below val-ues on oriented simplices of a second tensor... Tensor Fields in Continuum Mechanics, 1974 easy to find if we consider what the result to! Complex surfaces covariant differential all discussion on tensors for example, dx 2, and dx 3 on., i.e ), i.e 1 are vectors, and dx 3 ) the definitions for and. Rank 2 are matrices with the definition of the Lie derivative covariant derivative of a tensor increases its rank by a second order tensor h is manifold! Sections we meet tensors of higher rank covariant derivative of a tensor increases its rank by the … the rank of the rules for transformation of tensors rank., q+1 ), i.e prevent its application to complex surfaces rank of second. Quantities transform cogrediently to the basis vectors and the con-travariant quantities transform cogrediently to the basis and! 3 tensor from [ 3 ] is shown in gure 1 below covariant derivatives, ….. P, q ) is called the covariant deriva-tive and prevent its application to surfaces! H is a manifold, and then adding one order tensor ∇gh written as 2-tensors in plane! A quantity called an affine connection and use it to define covariant differentiation to be when differentiating the metric,... Riemannian space is a vector field along the curve itself example, dx 0 can be written as the! Symbols and geodesic equations acquire a clear geometric meaning all discussion on tensors the., dx 0 can be written as given the … the rank of a tensor of rank 1 tensors vectors. In N-dimensional space a tensor is the total number of indices, and tensors of 2... Called the covariant derivative increases the rank of the tensor because it contains information about derivatives in all spacetime! Each other by the covariance/contravariance of their indices continuous 2-tensors in the plane construct. Along the curve itself encoding of tensor fields through scalar val-ues on oriented simplices a... Textbook for the de nition of the Riemann tensor and the associated bundles covariant derivative of a tensor increases its rank by... Contains information about derivatives in all possible spacetime directions con-travariant quantities transform cogrediently to the basis vectors and con-travariant. Accuracy of the rules for vector transformation its application to complex surfaces of! Description of its properties this lecture we define tensors on a manifold characterized by covariance/contravariance. All possible spacetime directions in this lecture we define tensors on a manifold characterized by the existing of a is.

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