In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. The transformation...

and a covariant vector is a list of numbers that transforms in the same way. Contravariant vectors are often just called vectors and covariant vectors...

derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation. This article presents...

general covariant transformations are symmetries of gravitation theory on a world manifold X {\displaystyle X} . They are gauge transformations whose parameter...

some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is...

such gauge transformations, because they depend on the local frame. However, when gauge transformations act on fields and the gauge covariant derivative...

In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that...

contravariance Covariant derivative Fictitious force Galilean invariance Gauge covariant derivative General covariant transformations Harmonic coordinate...

linear transformations. Covariance and contravariance of vectors, properties of how vector coordinates change under a change of basis Covariant transformation...

one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field. The...

In particular, a Lorentz covariant scalar (e.g., the space-time interval) remains the same under Lorentz transformations and is said to be a Lorentz...

linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is...

\mathbf {F} } transforms covariantly. Not all gauge transformations can be generated by infinitesimal gauge transformations in general. An example is...

vector is called covariant or contravariant depending on how the transformation of the vector's components is related to the transformation of coordinates...

components from one such basis to another is done through an orthogonal transformation. The most familiar coordinate systems are the two-dimensional and three-dimensional...

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations...

G^{\text{op}}} are thus "turned around". Forming the opposite group becomes a (covariant) functor from Grp {\displaystyle {\textbf {Grp}}} to Grp {\displaystyle...

In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion...

called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a covariant functor...

internal symmetries encountered a problem of treating general covariant transformations and establishing the gauge status of a pseudo-Riemannian metric...

matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent...

on a principal frame bundle whose gauge symmetries are general covariant transformations which are not elements of a gauge group. In the physical literature...

are used to label a variable object as covariant (lower index), contravariant (upper index), or mixed covariant and contravariant (having both upper and...

components, Si j the mixed right-covariant components, Si j the mixed left-covariant components, and Sij the covariant components of the second-order tensor...

however, not Lorentz covariant. If a Lorentz transformation to a new inertial frame is carried out, a further gauge transformation has to be made to retain...

^{i}}_{jk}={\Gamma ^{i}}_{kj}.} The index-less transformation properties of a tensor are given by pullbacks for covariant indices, and pushforwards for contravariant...

A = H o m ( A , − ) {\displaystyle h_{A}=\mathrm {Hom} (A,-)} . The (covariant) hom-functor h A {\displaystyle h_{A}} sends X ∈ C {\displaystyle X\in...

position r and time t. Matrix exponentiation is used in the transformation. The gauge covariant derivative transforms similarly. The functions θn here are...

coordinates. More generally, the covariant components of a 4-vector transform according to the inverse Lorentz transformation: T μ ′ = Λ μ ′ ν T ν , {\displaystyle...

quantities must transform covariantly, that is, under a certain representation of the group of coordinate transformations between admissible frames of...