covariant differentiation is simply partial differentiation: ϕ ; a ≡ ∂ a ϕ {\displaystyle \phi _{;a}\equiv \partial _{a}\phi } For a contravariant vector...

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

the words covariant and contravariant refer to how objects transform under general coordinate transformations. Both covariant and contravariant four-vectors...

raised or lowered while retaining the base symbol, covariant indices are sometimes placed below contravariant indices for notational convenience (e.g. with...

transforms covariantly. The operation of associating to the (contravariant) components of a vector field v[f] = [ v1[f] v2[f] ... vn[f] ]T the (covariant) components...

a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index....

illustrate how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought...

to note that explicitly. Kronecker delta Levi-Civita symbol Covariant tensor Contravariant tensor The classical interpretation is by components. For example...

f : x → y and g : y → z, F(g ∘ f) = F(g) ∘ F(f). A contravariant functor F: C → D is like a covariant functor, except that it "turns morphisms around" ("reverses...

p_{x},p_{y},p_{z}\right),} τ the proper time of the particle, Fαβ the contravariant electromagnetic tensor F α β = ( 0 − E x / c − E y / c − E z / c E x...

In other terms, Θ is a tensor of type (1, 2) (carrying one contravariant and two covariant indices). Alternatively, the solder form can be characterized...

{\displaystyle Z_{ij}} and Z i j {\displaystyle Z^{ij}} are covariant and contravariant metric tensors, δ j i {\displaystyle \delta _{j}^{i}} is the...

matrix. The four-gradient of a scalar field φ transforms covariantly rather than contravariantly: ( 1 c ∂ ϕ ∂ t ′ ∂ ϕ ∂ x ′ ∂ ϕ ∂ y ′ ∂ ϕ ∂ z ′ ) = ( 1...

{e} ^{i}=\mathrm {d} x^{i}} refer to the unnormalized local covariant and contravariant bases respectively, g i j {\displaystyle g^{ij}} is the inverse...

This can be thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two T {\displaystyle {\mathcal {T}}} 's are...

Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively. The reciprocal lattice is the set of all...

Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf...

transform in a more general way which depends on the number of covariant indices and contravariant indices. spinor fields (such as the Dirac spinor) arise in...

in the technical sense: if  f : M → N is a smooth map and Ωk is the contravariant smooth functor that assigns to each manifold the space of k-forms on...

to any morphism of schemes. A bivariant theory is a pair of covariant and contravariant functors that assign to a map a group and a ring respectively...

as length contraction and time dilation and some relations to covariant and contravariant vectors were demonstrated by them. Gruner extended this method...

'lifting', known in special cases, and so is covariant (whereas a restricted representation is contravariant). Attempts to specify a direct construction...

Goldstein (1980) considers a passive (contravariant, or "alias") transformation, rather than the active (covariant, or "alibi") transformation here. His...

categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories – see § Categorical considerations. A topological...

{\frac {\mathrm {d} \xi ^{a}}{\mathrm {d} t}},} where Fa is the a-th contravariant component of the resultant force acting on the particle, Γabc are the...

discrete category { A }. contravariant functor A contravariant functor F from a category C to a category D is a (covariant) functor from Cop to D. It...

vaunt varius vari- varying bivariate, contravariance, contravariant, countervair, covariance, covariant, covariate, covariation, intervarietal, invariable...

either Ω or J. However, they have a different meaning: J is contravariant and Ω is covariant. The matrix Ω corresponds to the Lagrange brackets of classical...

or Sαβgαβ = 0 depending on the presentation of the field: covariant, mixed, and contravariant respectively. This doesn't necessarily come symmetric directly...