• covariant differentiation is simply partial differentiation: ϕ ; a ≡ ∂ a ϕ {\displaystyle \phi _{;a}\equiv \partial _{a}\phi } For a contravariant vector...
    36 KB (6,354 words) - 20:38, 2 May 2024
  • to label a variable object as covariant (lower index), contravariant (upper index), or mixed covariant and contravariant (having both upper and lower indexes)...
    13 KB (1,906 words) - 14:52, 10 February 2024
  • the words covariant and contravariant refer to how objects transform under general coordinate transformations. Both covariant and contravariant four-vectors...
    21 KB (2,917 words) - 02:30, 14 May 2024
  • 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 (8,861 words) - 18:17, 26 April 2024
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    a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index....
    69 KB (9,356 words) - 22:51, 9 May 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
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    illustrate how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought...
    80 KB (10,536 words) - 17:19, 14 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
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    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...
    57 KB (8,238 words) - 23:06, 31 January 2024
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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...
    9 KB (1,499 words) - 21:40, 1 November 2023