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a four-tensor with contravariant rank 1 and covariant rank 0. Four-tensors of this kind are usually known as four-vectors . Here the component x 0 = ct gives the displacement of a body in time (coordinate time t is multiplied by the speed of light c so that x 0 has dimensions of length).
Matrix rank should not be confused with tensor order, which is called tensor rank. Tensor order is the number of indices required to write a tensor , and thus matrices all have tensor order 2. More precisely, matrices are tensors of type (1,1), having one row index and one column index, also called covariant order 1 and contravariant order 1 ...
On the other hand, a randomly sampled complex tensor of the same size will be a rank-1 tensor with probability zero, a rank-2 tensor with probability one, and a rank-3 tensor with probability zero. It is even known that the generic rank-3 real tensor in R 2 ⊗ R 2 ⊗ R 2 {\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb ...
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 the mathematical theory of elasticity, Saint-Venant's compatibility condition defines the relationship between the strain and a displacement field by = (+) where ,. Barré de Saint-Venant derived the compatibility condition for an arbitrary symmetric second rank tensor field to be of this form, this has now been generalized to higher rank symmetric tensor fields on spaces of dimension
The total number of indices is also called the order, degree or rank of a tensor, [2] [3] [4] although the term "rank" generally has another meaning in the context of matrices and tensors. Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation.
Hooke's law has a symmetric fourth-order stiffness tensor with 81 components (3×3×3×3), but because the application of such a rank-4 tensor to a symmetric rank-2 tensor must yield another symmetric rank-2 tensor, not all of the 81 elements are independent. Voigt notation enables such a rank-4 tensor to be represented by a 6×6 matrix ...
The 4-divergence of the stress–energy tensor as the conserved Noether current associated with spacetime translations, gives four conservation laws in SR: [4]: 101–106 The conservation of energy (temporal direction) and the conservation of linear momentum (3 separate spatial directions).