8.13.1 Tensor Index Upper Position

Tensor Index Upper Position denotes the placement of indices in tensor notation, indicating contravariant components and their role in coordinate transformations.

Tensor Index Upper Position is the placement of a tensor index as a superscript, written above and to the right of the base symbol, used to denote a contravariant component or contravariant slot of a tensor. An index written in the upper position, such as the $i$ in $A^i$, transforms under a change of basis with the inverse of the matrix used to transform the basis vectors themselves, and it is this inverse transformation behavior that the upper position exists to encode. As with the lower position, the placement is a substantive statement about transformation law, not a typographic preference.

Meaning of the Upper Position

Contravariant Transformation Behavior

A quantity carrying an index in the upper position transforms with the inverse of whatever coefficients relate the old basis to the new one. Under a coordinate change, the components of a contravariant object obey

A^{i′} = \frac{\partial x^{i′}}{\partial x^{j}} A^{j}

The Jacobian factor here is inverse in role to the one used for lower-position indices: while a lower index picks up $\partial x^{j}/\partial x^{i'}$, an upper index picks up $\partial x^{i'}/\partial x^{j}$, and the two factors are mutual inverses of one another as matrices.

Association With Ordinary Vectors

The upper position is conventionally reserved for the components of ordinary vectors, also called contravariant vectors, such as displacement, velocity, or the coordinate differentials $dx^{i}$ themselves. This naming reflects a historical convention: coordinate differentials transform inversely (contra) to the way the basis vectors transform, which is exactly the behavior an upper index is defined to track.

Upper Position in Higher-Rank Tensors

Fully Contravariant Tensors

A tensor with every index in the upper position, such as $T^{ij}$ or $T^{ijk}$, is called a fully contravariant tensor of the corresponding rank. Each upper index transforms independently according to the contravariant law, so a rank-2 fully contravariant tensor obeys

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

Upper Indices Within Mixed Tensors

In a mixed tensor such as $T^{i}{}_{j}$, the upper index $i$ transforms contravariantly regardless of what happens to the lower index $j$ in the same expression; each index's transformation law is determined entirely by its own position, independent of the others attached to the same symbol.

Producing an Upper-Position Index From a Lower One

Index Raising via the Metric Tensor

A lower-position index can be converted into an upper-position index by contracting it with the contravariant metric tensor $g^{ij}$, the matrix inverse of the covariant metric:

A^{i} = g^{ij} A_{j}

This operation, called raising the index, requires a metric to be defined on the space, since without one there is no canonical way to identify covectors with vectors. Raising an index that was itself obtained by lowering an upper index, using the same metric, returns the original component unchanged, since $g^{ik}g_{kj} = \delta^{i}_{j}$.

Metric Inverse Relationship

The contravariant metric $g^{ij}$ used for raising and the covariant metric $g_{ij}$ used for lowering are matrix inverses of one another, satisfying $g^{ik}g_{kj} = \delta^{i}{j}$, where $\delta^{i}{j}$ is the Kronecker delta. This inverse relationship guarantees that raising and lowering are mutually cancelling operations.

Distinguishing the Upper Position From Exponent Notation

Avoiding Confusion With Powers

A superscript in ordinary algebra typically denotes exponentiation, as in $x^2$ meaning $x$ multiplied by itself. Within tensor notation, a superscript attached to an indexed symbol denotes an index label, not a power, and the surrounding context of tensor algebra is what signals this reading. To avoid genuine ambiguity, expressions requiring an actual power of a tensor component are conventionally written with parentheses around the base, such as $(A^{i})^{2}$, reserving the unparenthesized superscript exclusively for index notation.

Role Within Index Position Notation

The upper position, together with the lower position, completes the two-symbol vocabulary by which every tensor index is classified. The count of upper indices $p$ and lower indices $q$ on a given tensor determines its type $(p,q)$, and this type governs how the tensor combines with others under addition, multiplication, and contraction. The upper position specifically marks the contravariant slots of a tensor — the slots that pair naturally with covectors to produce scalars — making it one of the two foundational classifiers in all of tensor index notation.