10.18.3 Tensorial Rule Contravariant Factor Placement

The Tensorial Rule dictates how contravariant factors are positioned in tensor algebra, essential for maintaining coordinate independence in mathematical formulations.

Tensorial Rule Contravariant Factor Placement is the specific requirement, within the tensorial transformation rule, that every upper index of a tensor be paired with a forward Jacobian factor rather than an inverse Jacobian factor, and that this pairing be positioned consistently so that the free upper index of the resulting new component matches the free upper index carried by the forward Jacobian factor itself.

Statement of the Placement Rule

One Forward Jacobian per Upper Index

For each upper index carried by a tensor, exactly one forward Jacobian factor is inserted into the transformation formula, contracted against the corresponding original upper index through a summed dummy variable:

{\bar{V}}^{j} = \sum_{i}^{n} J_{i}^{j} V^{i}

Here the free upper index $j$ on the left-hand side matches exactly the free upper index on the forward Jacobian factor, while the summed index $i$ appears as a lower index on the forward Jacobian and as an upper index on the original component, satisfying the general index placement convention for a valid contraction.

Placement for Multiple Upper Indices

When a tensor carries several upper indices, each one independently receives its own forward Jacobian factor, with its own free index matching the corresponding position in the final result, and its own dummy index summed against the matching position of the original tensor:

{\bar{T}}^{kl} = \sum_{i}^{n} \sum_{j}^{n} J_{i}^{k} J_{j}^{l} T^{ij}

with each forward Jacobian factor independent of the other, so no cross-terms mixing the two upper indices through a single factor are allowed under the placement rule.

Why This Placement, Not the Inverse Jacobian

Consistency With Primary Basis Change

A contravariant index is associated with ordinary vectors expanded in the primary coordinate basis, and that primary basis itself changes according to the forward Jacobian under a passive change of basis; requiring a contravariant component to transform with the same forward Jacobian, rather than its inverse, is precisely what is needed to make the invariant sum of components times basis vectors come out consistently in both coordinate systems.

Consequence for Contracted Invariance

Placing the forward Jacobian on every upper index, paired with the inverse Jacobian on every lower index elsewhere in a contraction, is exactly the combination that collapses to the Kronecker delta through the Jacobian product identity when a full contraction is formed, guaranteeing that fully contracted scalars remain numerically invariant across the change of basis.

Diagram of Contravariant Placement

Upper Index Bound to Forward Jacobian

Placement in Mixed-Index Tensors

Independence From Lower Index Placement

In a mixed tensor, the contravariant factor placement rule applies exclusively to the upper indices and operates entirely independently of whatever placement rule governs the lower indices, so a mixed tensor's transformation formula simply combines the contravariant placement pattern for its upper indices with the covariant placement pattern for its lower indices, without any interaction between the two:

{\bar{T}}_{l}^{k} = \sum_{i}^{n} \sum_{j}^{n} J_{i}^{k} {(J^{- 1})}_{l}^{j} T_{j}^{i}

Order of Factors Does Not Affect the Result

Because the forward Jacobian factor for the upper index and the inverse Jacobian factor for the lower index act on entirely separate summed indices in a mixed tensor, they may be written in either order in the product without changing the value of the resulting sum, since ordinary multiplication of the numerical factors is commutative, even though the two factors remain conceptually distinct in what they represent.

Common Placement Errors

Swapping Forward and Inverse Jacobians

The most frequent error in applying the placement rule is attaching an inverse Jacobian factor to an upper index, or a forward Jacobian factor to a lower index, which produces a formula that fails to preserve contracted scalars and therefore does not describe a genuine tensorial transformation, even though the resulting expression may still look superficially similar to the correct formula.

Mismatched Free Index Labels

A second common error places the forward Jacobian's free index at a different position than the corresponding upper index on the left-hand side, producing an expression where the free indices no longer match across the equation, a violation of the basic index consistency requirement that applies to every tensorial formula regardless of variance type.