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  2. 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 ...

  3. Tensors in curvilinear coordinates - Wikipedia

    en.wikipedia.org/wiki/Tensors_in_curvilinear...

    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.

  4. Ricci calculus - Wikipedia

    en.wikipedia.org/wiki/Ricci_calculus

    [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]

  5. Glossary of tensor theory - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_tensor_theory

    Tensor algebra In the tensor algebra T(V) of a vector space V, the operation becomes a normal (internal) binary operation. A consequence is that T(V) has infinite dimension unless V has dimension 0. The free algebra on a set X is for practical purposes the same as the tensor algebra on the vector space with X as basis.

  6. 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.

  7. Category:Tensors - Wikipedia

    en.wikipedia.org/wiki/Category:Tensors

    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 .

  8. Invariants of tensors - Wikipedia

    en.wikipedia.org/wiki/Invariants_of_tensors

    A real tensor in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part.

  9. Poincaré–Birkhoff–Witt theorem - Wikipedia

    en.wikipedia.org/wiki/Poincaré–Birkhoff–Witt...

    The map L → U(L) of K-modules canonically extends to a map T(L) → U(L) of algebras, where T(L) is the tensor algebra on L (for example, by the universal property of tensor algebras), and this is a filtered map equipping T(L) with the filtration putting L in degree one (actually, T(L) is graded).