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The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner ...
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. For example, doing rotations over axis does not affect at all the properties of tensors, if a transformation law is followed.
The stress–energy tensor of a continuum or field generally takes the form of a second-order tensor, and usually denoted by T. The timelike component corresponds to energy density (energy per unit volume), the mixed spacetime components to momentum density (momentum per unit volume), and the purely spacelike parts to the 3d stress tensor.
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Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity, [8] in the mechanics of curved shells, [6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials [9] [10] and in many other fields.
A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b.Analogous to vectors, it can be written as a linear combination of the tensor basis e x ⊗ e x ≡ e xx, e x ⊗ e y ≡ e xy, ..., e z ⊗ e z ≡ e zz (the right-hand side of each identity is only an abbreviation, nothing more):
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