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...

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...

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

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...

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...

{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...