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

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

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

brackets enclose contravariant indices – the rule applies only to all enclosed contravariant indices, not to intermediately placed covariant indices. Parentheses...

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...

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...

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

Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively. The Brillouin zone is a Wigner–Seitz cell...

{\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...

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...

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...

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...

\mathbf {x} /\partial x^{i}} refer to the unnormalized local covariant and contravariant bases respectively, g i j {\displaystyle g^{ij}} is the inverse...

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

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...

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

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...

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...

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...

(D)} denotes the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called...

vector because of this, including Gibbs vectors, pseudovectors and contravariant vectors. The latter of these two, in plane-based GA, map to the concepts...

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

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

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

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...

{\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...

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...