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For example, in a fixed basis, a standard linear map that maps a vector to a vector, is represented by a matrix (a 2-dimensional array), and therefore is a 2nd-order tensor. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. Scalars are simple numbers and are thus 0th-order tensors.
The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix, [3] and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often very difficult to determine, and low rank decompositions of tensors are sometimes of great practical interest. [4]
Some useful relations in the calculus of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, [2] Simmonds, [4] Green and Zerna, [1] Basar and Weichert, [5] and Ciarlet. [6]
Order of a tensor The components of a tensor with respect to a basis is an indexed array. The order of a tensor is the number of indices needed. Some texts may refer to the tensor order using the term degree or rank. Rank of a tensor The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor.
This technique was first introduced into isotropic turbulence by Howard P. Robertson in 1940 where he was able to derive Kármán–Howarth equation from the invariant principle. [7] George Batchelor and Subrahmanyan Chandrasekhar exploited this technique and developed an extended treatment for axisymmetric turbulence.
Let M be a manifold, for instance the Euclidean plane R n.. Definition. A tensor field of type (p, q) is a section (, ()) where V is a vector bundle on M, V * is its dual and ⊗ is the tensor product of vector bundles.
The first-order Rivlin–Ericksen is given by = + where ... -th order Rivlin–Ericksen tensor. Higher-order tensor may be found iteratively by the expression (+ ...
The first index i indicates that the stress acts on a plane normal to the X i-axis, and the second index j denotes the direction in which the stress acts (For example, σ 12 implies that the stress is acting on the plane that is normal to the 1 st axis i.e.;X 1 and acts along the 2 nd axis i.e.;X 2). A stress component is positive if it acts in ...