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A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.
In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector ...
[a] [1] [2] [3] It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), tensor calculus or tensor analysis developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. [4]
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T • (V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property ...
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor [1]) is a tensor that can be written as a product of tensors of the form = where a, b, ..., d are nonzero and in V or V ∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors.
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):
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
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.