covariant differentiation is simply partial differentiation: ϕ ; a ≡ ∂ a ϕ {\displaystyle \phi _{;a}\equiv \partial _{a}\phi } For a contravariant vector...
37 KB (6,473 words) - 00:23, 24 September 2024
to label a variable object as covariant (lower index), contravariant (upper index), or mixed covariant and contravariant (having both upper and lower indexes)...
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Lorentz covariance (redirect from Lorentz covariant)
the words covariant and contravariant refer to how objects transform under general coordinate transformations. Both covariant and contravariant four-vectors...
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Metric tensor (redirect from Covariant metric tensor)
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...
56 KB (8,866 words) - 08:52, 9 August 2024
a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index....
69 KB (9,351 words) - 11:28, 12 October 2024
raised or lowered while retaining the base symbol, covariant indices are sometimes placed below contravariant indices for notational convenience (e.g. with...
43 KB (6,872 words) - 18:52, 6 May 2024
to note that explicitly. Kronecker delta Levi-Civita symbol Covariant tensor Contravariant tensor The classical interpretation is by components. For example...
8 KB (1,028 words) - 17:44, 11 January 2023
illustrate how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought...
79 KB (10,620 words) - 19:46, 7 October 2024
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...
34 KB (3,827 words) - 21:20, 10 October 2024
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...
60 KB (8,548 words) - 21:18, 7 September 2024
{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...
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{\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...
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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...
188 KB (25,173 words) - 14:53, 12 October 2024
Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively. The reciprocal lattice is the set of all...
32 KB (5,485 words) - 22:43, 6 September 2024
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...
27 KB (4,357 words) - 11:40, 1 October 2024
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...
31 KB (4,234 words) - 03:54, 23 March 2024
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...
33 KB (4,371 words) - 12:42, 10 October 2024
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...
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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...
34 KB (4,033 words) - 15:22, 7 October 2024
'lifting', known in special cases, and so is covariant (whereas a restricted representation is contravariant). Attempts to specify a direct construction...
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Goldstein (1980) considers a passive (contravariant, or "alias") transformation, rather than the active (covariant, or "alibi") transformation here. His...
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categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories – see § Categorical considerations. A topological...
39 KB (5,806 words) - 03:42, 8 May 2024
as length contraction and time dilation and some relations to covariant and contravariant vectors were demonstrated by them. Gruner extended this method...
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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...
26 KB (4,195 words) - 23:25, 28 November 2023
{\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...
92 KB (14,503 words) - 13:33, 9 October 2024
vaunt varius vari- varying bivariate, contravariance, contravariant, countervair, covariance, covariant, covariate, covariation, intervarietal, invariable...
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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...
73 KB (11,111 words) - 23:21, 4 October 2024
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...
21 KB (3,267 words) - 21:16, 17 October 2023
or Sαβgαβ = 0 depending on the presentation of the field: covariant, mixed, and contravariant respectively. This doesn't necessarily come symmetric directly...
149 KB (19,750 words) - 15:33, 21 June 2024