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A more general notation is tensor index notation, which has the flexibility of numerical values rather than fixed coordinate labels. The Cartesian labels are replaced by tensor indices in the basis vectors e x ↦ e 1, e y ↦ e 2, e z ↦ e 3 and coordinates a x ↦ a 1, a y ↦ a 2, a z ↦ a 3.
Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, [2] Naghdi, [3] Simmonds, [4] Green and Zerna, [1] Basar and Weichert, [5] and Ciarlet. [6]
The earliest foundation of tensor theory – tensor index notation. [1] 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 ...
A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called a permutation tensor. Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure rotations, consistent with that it is (by definition) the same in all coordinate systems ...
In Ricci's notation, he refers to "systems" with covariant and contravariant components, which are known as tensor fields in the modern sense. [16] In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Albert Einstein's theory of general relativity, around 1915. General ...
In mathematics, a tensor is a certain kind of geometrical entity and array concept. It generalizes the concepts of scalar , vector and linear operator , in a way that is independent of any chosen frame of reference .
In general, a tensor operator is one that transforms according to a tensor: † ^ =,,, ^,,, where the basis are transformed by or the vector components transform by . In the subsequent discussion surrounding tensor operators, the index notation regarding covariant/contravariant behavior is ignored entirely.
In a Cartesian coordinate system we have the following relations for a vector ... is the fourth order identity tensor. In index notation with respect to an ...