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...

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

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...

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

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

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...

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

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

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

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

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...

\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...

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

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...

→ C o v a r i a n t {\displaystyle \mathrm {Covariant} \circ \mathrm {Covariant} \to \mathrm {Covariant} } C o n t r a v a r i a n t ∘ C o n t r a v...

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...

summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a passive transformation between bases, with the components...

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

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

(x)} but can also be derived by asking that covariant derivatives transform under a gauge transformation as D μ ψ ( x ) ↦ e i α ( x ) D μ ψ ( x ) , {\displaystyle...

convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form. Maxwell's equations, when they were first stated in their...

}\varphi +kB_{\mu }\varphi } is covariant and has conformal weight k − 1 {\displaystyle k-1} . For the transformation g a b = f ( ϕ ( x ) ) g ¯ a b {\displaystyle...

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

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...

Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative Exterior derivative Exterior product Hodge star...

relationship between different systems is described by coordinate transformations, which give formulas for the coordinates in one system in terms of...

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...