10.18.2 Tensorial Rule Covariant Factor Placement

The Tensorial Rule ensures covariant factor placement in tensor algebra, maintaining invariance under coordinate transformations.

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

Statement of the Placement Rule

One Inverse Jacobian per Lower Index

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

{\bar{W}}_{j} = \sum_{i}^{n} {(J^{- 1})}_{j}^{i} W_{i}

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

Placement for Multiple Lower Indices

When a tensor carries several lower indices, each one independently receives its own inverse 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^{- 1})}_{k}^{i} {(J^{- 1})}_{l}^{j} T_{ij}

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

Why This Placement, Not the Forward Jacobian

Consistency With Dual Basis Change

A covariant index is, by definition, associated with the dual basis, and the dual basis itself changes according to the inverse Jacobian under a passive basis change, so requiring a covariant component to transform with the inverse Jacobian is exactly what is needed to keep the pairing between a covector and its dual basis invariant across the change of basis.

Consequence for Contracted Invariance

Placing the inverse Jacobian on every lower index, and the forward Jacobian on every upper index, is precisely the placement that causes a full contraction between an upper and a lower index to collapse to the Kronecker delta through the Jacobian product identity, which is what guarantees the invariance of fully contracted scalars; any other placement would break this cancellation.

Diagram of Covariant Placement

Lower Index Bound to Inverse Jacobian

Placement in Mixed-Index Tensors

Independence From Upper Index Placement

In a mixed tensor, the covariant factor placement rule applies exclusively to the lower indices and operates entirely independently of whatever placement rule governs the upper indices, so a mixed tensor's transformation formula simply combines the covariant placement pattern for its lower indices with the contravariant placement pattern for its upper 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 and inverse Jacobian factor 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 involved is commutative, even though the two factors are 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 a forward Jacobian factor to a lower index, or an inverse Jacobian factor to an upper 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 inverse Jacobian's free index at a different position than the corresponding lower 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.