a differential graded algebra A over a commutative ring R, the derived tensor product functor is − ⊗ A L − : D ( M A ) × D ( A M ) → D ( R M ) {\displaystyle...

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

of differential operators. Tor functor Tensor product of algebras Tensor product of fields Derived tensor product Tensoring with M the exact sequence 0...

tensor"), may be analyzed either by artificial neural networks or tensor methods. Tensor decomposition factorizes data tensors into smaller tensors....

theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their...

individual left derived functors one can also take the total derived functor of the tensor functor. This gives rise to the derived tensor product − ⊗ L − :...

Tensor networks or tensor network states are a class of variational wave functions used in the study of many-body quantum systems and fluids. Tensor networks...

mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There...

{\displaystyle n} and a tensor of order m {\displaystyle m} is a tensor of order n + m − 2 {\displaystyle n+m-2} , see Tensor contraction for details...

factor is expressed in terms of the Tor functor, the first derived functor of the tensor product. When R is a PID, then the correct statement of the Künneth...

electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a...

of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application of tensor theory...

independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms...

In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central...

a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from...

manifold M {\displaystyle M} and the metric tensor is given as a covariant, second-degree, symmetric tensor on M {\displaystyle M} , conventionally denoted...

cross product can be seen as the (1,2)-tensor (a mixed tensor, specifically a bilinear map) obtained from the 3-dimensional volume form, a (0,3)-tensor, by...

, fiber product of immersions) does not yield the correct intersection number. In the derived context, one takes the derived tensor product A ⊗ L B {\displaystyle...

}_{i_{r+p}}.} The components of this tensor are precisely the skew part of the components of the tensor product s ⊗ t, denoted by square brackets on the...

deformation tensors. In 1839, George Green introduced a deformation tensor known as the right Cauchy–Green deformation tensor or Green's deformation tensor (the...

\otimes K_{s_{r}}} where ⊗ {\displaystyle \otimes } denotes the derived tensor product of chain complexes of A-modules. When s 1 , … , s r {\displaystyle...

consequently a vector is called a contravariant tensor. A vector, which is an example of a contravariant tensor, has components that transform inversely to...

inertia tensor of a body calculated at its center of mass, and R {\displaystyle \mathbf {R} } be the displacement vector of the body. The inertia tensor of...

spectra keeps track of the derived information of commutative rings, where the smash product acts as the derived tensor product. Moreover, Eilenberg–Maclane...

differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors...

differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If T is a tensor field and X is a vector field...

structure tensor is often used in image processing and computer vision. For a function I {\displaystyle I} of two variables p = (x, y), the structure tensor is...

^{L}K\right)} where Rp* is the derived direct image functor and ⊗ L {\displaystyle \otimes ^{L}} is the derived tensor product. Fourier-Mukai transforms always...

is locally given by a derived tensor product) agrees with the ordinary pullback (locally given by the underived tensor product), and since the pullback...

simplest tensor quantities, which are tensors can be used to describe more general physical properties. For example, the Cauchy stress tensor possesses...