6.7.2 Tensor Type Covariant Part

Tensor Type Covariant Part is the sub-structure of a type (p, q) tensor's domain corresponding specifically to its q covariant slots, isolating the factor V^{⊗q} (or, dually, the q-fold product of V accepted by the map) from the full domain (V*)^{⊗p} ⊗ V^{⊗q} and treating it as a distinct object whose transformation behavior, dimension count, and algebraic role can be studied on its own, separately from the contravariant part built from the remaining p slots. Isolating the covariant part this way reflects the fact that within type classification, the p and q slots transform by different rules and often play structurally different roles, even though both are counted together in the single pair (p, q).

Defining the Covariant Part

The Factor Corresponding to Lower Indices

For a type (p, q) tensor T_{j₁...j_q}^{i₁...i_p}, the covariant part is the portion of the index structure carried by the q lower indices j₁, ..., j_q; algebraically, it corresponds to the factor V^{⊗q} in the domain of the associated multilinear map, the part of the domain built from ordinary vectors rather than covectors.

with the second factor, ×_q V, identified as the covariant part of the domain.

The Covariant Part as a Sub-Tensor When p Slots Are Fixed

Fixing all p contravariant arguments to specific covectors reduces T to a purely q-linear map on V, T(ω₁, ..., ω_p, ·, ..., ·) : V × ... × V → F; this reduced map is exactly the covariant part of T made concrete once the contravariant part has been supplied with specific values.

Diagram Isolating the Covariant Factor

Transformation Behavior of the Covariant Part

The Direct-Jacobian Rule Applies Only Here

Under a change of basis x' = Jx, each of the q covariant slots transforms with the Jacobian directly:

a rule applied independently to each of the q covariant indices and never to any of the p contravariant indices, which follow the separate inverse-Jacobian rule instead; the covariant part is precisely the collection of slots to which this direct-Jacobian rule is confined.

Consistency Across All q Covariant Slots

Every one of the q covariant slots transforms by the identical direct-Jacobian rule; the covariant part does not further distinguish among its own q slots by any different transformation behavior, so within the covariant part itself, all slots are transformation-equivalent, differing (if at all) only in the specific position they occupy under valence classification.

Dimension and Component Count of the Covariant Part

Dimension of the Covariant Factor Alone

is the dimension contributed by the covariant part alone, entering the full component count dim(V)^{p+q} of the entire tensor as one of its two multiplicative factors, dim(V)^p · dim(V)^q.

Isolating the Covariant Part's Contribution to Total Components

Because the full component count factors as a product over the contravariant and covariant parts separately, doubling q while holding p fixed multiplies the total component count by dim(V)^q, an effect attributable entirely to the covariant part and independent of whatever the contravariant part happens to be.

The Covariant Part Under Algebraic Operations

Covariant Part Under the Tensor Product

The covariant part of A ⊗ B is the concatenation of the covariant part of A with the covariant part of B, exactly mirroring how the full type's q-component adds; the contravariant parts of the two factors similarly concatenate independently, with no mixing between contravariant and covariant parts across the two factors.

Covariant Part Under Contraction

A contraction pairs one contravariant slot against one covariant slot, so it always removes exactly one slot from the covariant part (reducing its exponent q by 1) together with exactly one slot from the contravariant part (reducing p by 1); the covariant part alone, considered in isolation, is never reduced by two slots at once, since contraction inherently spans both parts rather than acting within just one.

Why the Covariant Part Matters

Isolating the Half of a Mixed Tensor That Behaves Like an Ordinary Multilinear Form

Once the contravariant part is fixed, the covariant part alone behaves exactly like an ordinary q-linear form on V, allowing techniques developed for purely covariant tensors — such as symmetrization or antisymmetrization restricted to the covariant slots — to be applied directly to this part of a mixed tensor without needing to account for the contravariant part at all.

Supporting Separate Reasoning About Each Half of a Mixed Type

Because the covariant and contravariant parts transform independently and combine independently under the tensor product, many arguments about a mixed type (p, q) tensor can be split into a claim about the covariant part and a separate, symmetric claim about the contravariant part, simplifying proofs that would otherwise need to track both kinds of slot simultaneously.