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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.
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 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.
Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
A (0,1) tensor is a covector. A (0,2) tensor is a bilinear form. An example is the metric tensor . A (1,1) tensor is a linear map. An example is the delta, , which is the identity map, or a Lorentz transformation .
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry , algebraic geometry , general relativity , in the analysis of stress and strain in material object, and ...
A general tensor may be antisymmetrized or symmetrized, and there is according notation. We demonstrate the notation by example. Let's antisymmetrize the type-(0,3) tensor ω a b c {\displaystyle \omega _{abc}} , where S 3 {\displaystyle \mathrm {S} _{3}} is the symmetric group on three elements.
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