8.13.3 Tensor Index Mixed Position

Tensor Index Mixed Position combines upper and lower indices, representing contravariant and covariant components in tensor notation.

Tensor Index Mixed Position is the arrangement of indices on a tensor symbol in which at least one index occupies the upper (contravariant) position and at least one other index occupies the lower (covariant) position on the same symbol. A tensor exhibiting mixed position, such as $T^{i}{}_{j}$, is called a mixed tensor, and each of its indices retains the transformation law dictated purely by its own placement, independent of the positions of the other indices attached to the same object. Mixed position is the general case of tensor indexing, of which fully contravariant and fully covariant tensors are the two extremes.

Structure of Mixed-Position Tensors

Independent Transformation of Each Index

In a mixed tensor, every index transforms according to its own position without reference to the type of the neighboring indices. For $T^{i}{}_{j}$, the upper index transforms contravariantly and the lower index transforms covariantly, combining into the full transformation law

T_{j′}^{i′} = \frac{\partial x^{i′}}{\partial x^{k}} \frac{\partial x^{l}}{\partial x^{j′}} T_{l}^{k}

where the upper index picks up the contravariant Jacobian factor and the lower index picks up the covariant Jacobian factor, multiplied together for the combined component.

Notation for Multiple Mixed Indices

A tensor of type $(p,q)$ with $p$ upper and $q$ lower indices is written with all upper letters as superscripts and all lower letters as subscripts on the same base symbol, for example $T^{ij}{}{kl}$ for a $(2,2)$ tensor. Each superscript and each subscript independently follows the transformation rule appropriate to its own position, so the full transformation of $T^{ij}{}{kl}$ combines two contravariant Jacobian factors and two covariant Jacobian factors.

Staggered Notation for Ordered Mixed Indices

When Left-to-Right Order Matters

For tensors that are not symmetric across their upper and lower slots, the relative ordering of the upper and lower indices can carry meaning, and a staggered notation is used to preserve it, such as $T^{i}{}_{j}{}^{k}$. This placement records that the index $i$ occupies the first slot, $j$ the second, and $k$ the third, even though $i$ and $k$ are both upper while $j$ is lower.

Distinguishing Staggered Orderings

Because the physical or geometric meaning of a tensor's slots can depend on their original order of construction — for instance, when a mixed tensor arises from contracting several separate tensors together — $T^{i}{}{j}{}^{k}$ and $T^{ik}{}{j}$ may denote genuinely different objects unless the tensor is known to be symmetric under exchange of its upper indices. Staggered notation exists specifically to prevent this ambiguity from being silently introduced by regrouping the superscripts and subscripts.

The Kronecker Delta as the Canonical Mixed Tensor

A Type (1,1) Invariant Object

The Kronecker delta $\delta^{i}{}_{j}$ is the simplest and most fundamental mixed tensor, defined by the property that it equals 1 when $i = j$ and 0 otherwise, in every coordinate system. As a type $(1,1)$ tensor, it transforms as

δ_{j′}^{i′} = \frac{\partial x^{i′}}{\partial x^{k}} \frac{\partial x^{l}}{\partial x^{j′}} δ_{l}^{k}

and remains numerically identical in every basis precisely because the two Jacobian factors are inverse matrices of one another, so their product collapses back to the identity.

Role in Contraction

Contracting any tensor's upper or lower index against the Kronecker delta leaves that tensor unchanged, since $\delta^{i}{}_{j} A^{j} = A^{i}$; the mixed Kronecker delta functions as the identity operator with respect to implicit contraction, precisely because its indices are in mixed position.

Producing Mixed Position From Raising and Lowering

Partial Application of the Metric

Starting from a fully covariant or fully contravariant tensor, applying the metric to raise or lower only some of its indices produces a mixed-position tensor. For example, raising the first index of a fully covariant $T_{ij}$ gives the mixed tensor

T^{i}_{j} = g^{ik} T_{kj}

leaving the second index unaffected in the lower position while the first has been converted to the upper position.

Role Within Index Position Notation

Mixed position is the general framework that subsumes the two pure cases — fully covariant and fully contravariant tensors — as special instances in which $q = 0$ or $p = 0$, respectively. Since most tensors of interest beyond rank 1 arise from contractions, products, or metric operations applied to simpler objects, mixed position is in practice the most common configuration encountered in tensor algebra, and staggered notation together with independent per-index transformation rules is what allows such tensors to be manipulated correctly and unambiguously.