In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative...

tensor product of v {\displaystyle v} and w {\displaystyle w} . An element of V ⊗ W {\displaystyle V\otimes W} is a tensor, and the tensor product of...

leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors". Tullio Levi-Civita...

correspondence between quadratic forms and symmetric bilinear forms breaks down. By the universal property of the tensor product, there is a canonical...

differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional...

from the tensor algebra. See the article on tensor algebras for a detailed treatment of the topic. The exterior product of multilinear forms defined above...

F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor product...

antisymmetric covariant tensor field of rank k. The differential forms on M are in one-to-one correspondence with such tensor fields. As well as the addition...

A metric tensor g is positive-definite if g(v, v) > 0 for every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known...

the inner product. The inertia tensor in mechanics is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to...

vector space with a quadratic form. The Clifford algebra of U + V is isomorphic to the tensor product of the Clifford algebras of U and (−1)dim(U)/2dV...

topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see...

mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings;...

(alternatively, wedge product). The inner product is more correctly called a scalar product in this context, as the nondegenerate quadratic form in question need...

categorical traces in the abstract setting of category theory. Trace of a tensor with respect to a metric tensor Characteristic function Field trace Golden–Thompson...

mathematics, a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields....

graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which...

a basis for the cotangent space at p. The tensor product (denoted by the symbol ⊗) yields a tensor field of type (0, 2), i.e. the type that expects two...

vector, and tensor forms. Without reference to any of these particularities , let f {\displaystyle f} be a square-integrable function of physical space...

vector space to V . {\displaystyle V.} By the universal property of tensor products these are in one-to-one correspondence with complex linear maps V...

The Witt group of k can be given a commutative ring structure, by using the tensor product of quadratic forms to define the ring product. This is sometimes...

on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal...

\otimes } is the outer product. Let R {\displaystyle \mathbf {R} } be the matrix that represents a body's rotation. The inertia tensor of the rotated body is...

mathematics, the signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional...

associated to this inner product. An example is provided by the Hilbert space L2([0, 1]). The Hilbertian tensor product of two copies of L2([0, 1]) is isometrically...

the notion of ε-quadratic forms and ε-symmetric forms. In terms of representation theory: exchanging variables gives a representation of the symmetric...

a vector space X with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps X → X for...

number of real roots and 2r2 is the number of non-real complex roots of f (which come in complex conjugate pairs); write the tensor product of fields...

space equipped with a quadratic form. Over the real numbers this is equivalent to being able to define a symmetric scalar product, u ⋅ v = ⁠1/2⁠(uv + vu)...

(nondegenerate) quadratic form, such as Euclidean space with its standard dot product or Minkowski space with its standard Lorentz metric. The space of spinors...