completely antisymmetric contravariant tensor field may be referred to as a k {\displaystyle k} -vector field. A tensor A that is antisymmetric on indices...

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, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T (...

that B(x, y) = −B(y, x) for all x and y. Antisymmetric tensor in matrices and index subsets. "antisymmetric function" – odd function Symmetry in mathematics...

essentially the composition of functions. In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting...

notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern...

or alternatively as the exterior product to obtain a second order antisymmetric tensor x ∧ p. What does this combine with, if anything? There is another...

calculus on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors. A tensor bundle is a...

to get a maximally covariant antisymmetric tensor. Raising an index on an (n, m)-tensor produces an (n + 1, m − 1)-tensor; this corresponds to moving diagonally...

be an antisymmetric tensor of rank ⁠ r {\displaystyle r} ⁠. Then, for α ∈ V∗, ⁠ ι α t {\displaystyle \iota _{\alpha }t} ⁠ is an alternating tensor of rank...

limitations of vector calculus; on the other hand, when expressed as an antisymmetric tensor field via the wedge operator of geometric calculus, the curl generalizes...

The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field...

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

tensor field is found to satisfy the equations of a Maxwell–Proca massive antisymmetric tensor field. This led Moffat to propose metric-skew-tensor-gravity...

torque, also second order antisymmetric tensor. The electromagnetic field tensor is another second order antisymmetric tensor field, with six components:...

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

pseudovectors and antisymmetric tensors of order 2. The dual of a pseudovector is an antisymmetric tensor of order 2 (and vice versa). The tensor is an invariant...

metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g(v, v) > 0 for...

The electromagnetic field tensor F a b {\displaystyle F^{ab}} , a rank-two antisymmetric tensor. Although the word 'tensor' refers to an object at a point...

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

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...

The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed...

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

differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing...

mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the...

and ♯ maps k-covectors to k-vectors, all the indices of a totally antisymmetric tensor are simultaneously raised or lowered, and so no index need be indicated:...

Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann...

thought of as a tensor, and is written δ j i {\displaystyle \delta _{j}^{i}} . Sometimes the Kronecker delta is called the substitution tensor. In the study...

relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of...

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...