6.7.3 Tensor Type Contravariant Part

The contravariant part of a tensor type describes how it transforms under coordinate changes, playing a key role in tensor algebra and differential geometry.

Tensor Type Contravariant Part is the sub-structure of a type (p, q) tensor's domain corresponding specifically to its p contravariant slots, isolating the factor (V*)^{⊗p} 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 covariant part built from the remaining q slots. Isolating the contravariant part this way mirrors exactly how the covariant part is isolated, reflecting that the two parts of a mixed tensor's domain transform by different rules and are naturally treated as separable components of the whole.

Defining the Contravariant Part

The Factor Corresponding to Upper Indices

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

T : \underset{p}{\times} V^{*} \times \underset{q}{\times} V \to F

with the first factor, ×_p V*, identified as the contravariant part of the domain.

The Contravariant Part as a Sub-Tensor When q Slots Are Fixed

Fixing all q covariant arguments to specific vectors reduces T to a purely p-linear map on V*, T(·, ..., ·, v₁, ..., v_q) : V* × ... × V* → F; this reduced map is exactly the contravariant part of T made concrete once the covariant part has been supplied with specific values.

Diagram Isolating the Contravariant Factor

Transformation Behavior of the Contravariant Part

The Inverse-Jacobian Rule Applies Only Here

Under a change of basis x' = Jx, each of the p contravariant slots transforms with the inverse Jacobian:

T^{i^{'}} = \sum_{i} J_{i}^{i^{'}} - 1 T^{i}

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

Consistency Across All p Contravariant Slots

Every one of the p contravariant slots transforms by the identical inverse-Jacobian rule; the contravariant part does not further distinguish among its own p slots by any different transformation behavior, so within the contravariant 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 Contravariant Part

Dimension of the Contravariant Factor Alone

\dim ({(V^{*})}^{\otimes p}) = \dim (V)^{p}

is the dimension contributed by the contravariant 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, using dim(V*) = dim(V) for finite-dimensional V.

Isolating the Contravariant Part's Contribution to Total Components

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

The Contravariant Part Under Algebraic Operations

Contravariant Part Under the Tensor Product

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

Contravariant Part Under Contraction

A contraction pairs one contravariant slot against one covariant slot, so it always removes exactly one slot from the contravariant part (reducing its exponent p by 1) together with exactly one slot from the covariant part (reducing q by 1); the contravariant 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 Contravariant Part Matters

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

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

Completing the Symmetric Picture Alongside the Covariant Part

Together with the covariant part, the contravariant part shows that a mixed type (p, q) tensor decomposes cleanly into two independently transforming, independently combining halves, confirming from the opposite side of the type pair that many arguments about a mixed tensor can be split into a claim about each part separately rather than requiring both kinds of slot to be tracked simultaneously throughout.