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  2. Tensor - Wikipedia

    en.wikipedia.org/wiki/Tensor

    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.

  3. Ricci calculus - Wikipedia

    en.wikipedia.org/wiki/Ricci_calculus

    While much of the notation may be applied with any tensors, operations relating to a differential structure are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays. A tensor may be expressed as a linear sum of the tensor product of vector and covector basis ...

  4. String diagram - Wikipedia

    en.wikipedia.org/wiki/String_diagram

    the tensor of two diagrams as the composition of whiskerings ′ = (′) ′. Note that because the diagram is in generic form (i.e. each layer contains exactly one box) the definition of tensor is necessarily biased: the diagram on the left hand-side comes above the one on the right-hand side.

  5. Glossary of tensor theory - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_tensor_theory

    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 ...

  6. Cartesian tensor - Wikipedia

    en.wikipedia.org/wiki/Cartesian_tensor

    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.

  7. Penrose graphical notation - Wikipedia

    en.wikipedia.org/wiki/Penrose_graphical_notation

    Penrose graphical notation (tensor diagram notation) of a matrix product state of five particles. In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. [1]

  8. Tensor algebra - Wikipedia

    en.wikipedia.org/wiki/Tensor_algebra

    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 ...

  9. Raising and lowering indices - Wikipedia

    en.wikipedia.org/wiki/Raising_and_lowering_indices

    Concretely, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by V ∗ := Hom ( V , K ) {\displaystyle V^{*}:={\text{Hom}}(V,K)} , so that α ∈ V ∗ {\displaystyle \alpha \in V^{*}} is a linear map α : V → K {\displaystyle \alpha :V\rightarrow K} .