** Metric tensor, in mathematics, a symmetric rank-2 tensor, used to measure length and angle

For the anisotropic material, it requires the mathematics of a second order tensor and up to 21 material property constants.

In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept.

In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space ( typically a Euclidean space or manifold ).

In mathematics, the Lie derivative (), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field ( including scalar function, vector field and one-form ), along the flow of another vector field.

The following is only an introduction to the concept: index notation is used in more detail in mathematics ( particularly in the representation and manipulation of tensor operations ).

* For the connection to mathematics, see curvature tensor,

In mathematics, a monoidal category ( or tensor category ) is a category C equipped with a bifunctor

In mathematics, the tensor algebra of a vector space V, denoted T ( V ) or T < sup >•</ sup >( V ), is the algebra of tensors on V ( of any rank ) with multiplication being the tensor product.

In mathematics, the tensor product of two R-algebras is also an R-algebra.

In other cases, specialist terms have been created which do not exist outside of mathematics — examples are tensor, fractal, functor.

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras.

In mathematics, tensor calculus or tensor analysis is an advanced extension of vector calculus to more general mathematical objects called tensors.

In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of O < sub > X </ sub >- modules.

In mathematics and physics, in particular differential geometry and general relativity, a warped geometry is a Riemannian or Lorentzian manifold whose metric tensor can be written in form

In mathematics and theoretical physics, a bifundamental representation is a representation obtained as a tensor product of two fundamental or antifundamental representations.

In mathematics and theoretical physics, a tensor is antisymmetric on ( or with respect to ) an index subset if it alternates sign when any two indices of the subset are interchanged.

Function rank is an important concept to array programming languages in general, by analogy to tensor rank in mathematics: functions that operate on data may be classified by the number of dimensions they act on.

It is a matter of pure mathematics that, in any metric theory, the Riemann tensor can always be written as the sum of the Weyl curvature ( or conformal curvature tensor ) plus a piece constructed from the Einstein tensor.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

In mathematics, one can often define a direct product of objects

In mathematics, an inner product space is a vector space with an additional structure called an inner product.

In mathematics, one can define a product of group subsets in a natural way.

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.

In mathematics, it is possible to combine several rings into one large product ring.

In mathematics, the word is used as a mnemonic for the cross product.

In mathematics, the Klein four-group ( or just Klein group or Vierergruppe (), often symbolized by the letter V ) is the group Z < sub > 2 </ sub > × Z < sub > 2 </ sub >, the direct product of two copies of the cyclic group of order 2.

Social constructivism or social realism theories see mathematics primarily as a social construct, as a product of culture, subject to correction and change.

In mathematics, specifically in group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup.

* Multiplicative inverse, in mathematics, the number 1 / x, which multiplied by x gives the product 1, also known as a reciprocal

In category theory and its applications to mathematics, a biproduct of a finite collection of objects in a category with zero object is both a product and a coproduct.

In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses ( disregarding trivial variations such as st < sup >− 1 </ sup > = su < sup >− 1 </ sup > ut < sup >− 1 </ sup >).

In mathematics, a unique factorization domain ( UFD ) is a commutative ring in which every non-unit element, with special exceptions, can be uniquely written as a product of prime elements ( or irreducible elements ), analogous to the fundamental theorem of arithmetic for the integers.

The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements.

In mathematics, factorization ( also factorisation in British English ) or factoring is the decomposition of an object ( for example, a number, a polynomial, or a matrix ) into a product of other objects, or factors, which when multiplied together give the original.

In mathematics, particularly linear algebra and numerical analysis, the Gram – Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space R < sup > n </ sup >.

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact.

In mathematics, the Lambert W function, also called the Omega function or product logarithm, is a set of functions,

In mathematics, the Hadamard product may refer to:

In some areas of mathematics, associative algebras are typically assumed to have a multiplicative unit, denoted 1.

An example of such a finite field is the ring Z / pZ, which is essentially the set of integers from 0 to p − 1 with integer addition and multiplication modulo p. It is also sometimes denoted Z < sub > p </ sub >, but within some areas of mathematics, particularly number theory, this may cause confusion because the same notation Z < sub > p </ sub > is used for the ring of p-adic integers.

The Julia set of a function ƒ is commonly denoted J ( ƒ ), and the Fatou set is denoted F ( ƒ ).< ref > Note that for other areas of mathematics the notation can also represent the Jacobian matrix of a real valued mapping between smooth manifolds .</ ref > These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century.

The n × n orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by O ( n ), which — with its subgroups — is widely used in mathematics and the physical sciences.

The cross product of two vectors a and b is denoted by In physics, sometimes the notation is used, though this is avoided in mathematics to avoid confusion with the exterior product.

In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp ( 2n, F ) and Sp ( n ).

In mathematics, the unitary group of degree n, denoted U ( n ), is the group of n × n unitary matrices, with the group operation that of matrix multiplication.

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1 / x or x < sup >− 1 </ sup >, is a number which when multiplied by x yields the multiplicative identity, 1.

In mathematics, specifically group theory, the index of a subgroup H in a group G is the " relative size " of H in G: equivalently, the number of " copies " ( cosets ) of H that fill up G. For example, if H has index 2 in G, then intuitively " half " of the elements of G lie in H. The index of H in G is usually denoted | G: H | or < nowiki ></ nowiki >.

* In the history of mathematics, these symbols have denoted numbers, shapes, patterns, and change.

In mathematics, a cube root of a number, denoted or x < sup > 1 / 3 </ sup >, is a number a such that a < sup > 3 </ sup > = x.

In mathematics, the constructible universe ( or Gödel's constructible universe ), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets.

* Special linear group, a term used in mathematics, denoted SL < sub > n </ sub >

In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic constant of the cube, denoted M < sub > 3 </ sub >( n ).

In mathematics, a magic tesseract is the 4-dimensional counterpart of a magic square and magic cube, that is, a number of integers arranged in an n × n × n × n pattern such that the sum of the numbers on each pillar ( along any axis ) as well as the main space diagonals is equal to a single number, the so-called magic constant of the tesseract, denoted M < sub > 4 </ sub >( n ).

In mathematics, a magic hypercube is the k-dimensional generalization of magic squares, magic cubes and magic tesseracts ; that is, a number of integers arranged in an n × n × n × ... × n pattern such that the sum of the numbers on each pillar ( along any axis ) as well as the main space diagonals is equal to a single number, the so-called magic constant of the hypercube, denoted M < sub > k </ sub >( n ).

In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ (·).

* the spectrum of a matrix, denoted as, in applied mathematics

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps.

In mathematics, the axiom of dependent choices, denoted DC, is a weak form of the axiom of choice ( AC ) which is still sufficient to develop most of real analysis.

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets.

In mathematics, the category of magmas, denoted Mag, has as objects sets with a binary operation, and morphisms given by homomorphisms of operations ( in the universal algebra sense ).