7.22.2 Tensor Component Superscript Index

The Tensor Component Superscript Index denotes the position of a component in a tensor, specifying its transformation properties and role in multi-linear algebra.

Tensor Component Superscript Index is the upper index attached to a tensor component symbol, such as the i in Tᵢ, that identifies the component as transforming contravariantly under a change of basis or coordinates. A superscript index marks a slot of the tensor that is dual to a covector slot: it is the index carried by the tensor's vector-like (contravariant) arguments, and it is precisely the index type that a covariant (subscript) index is built to pair with and cancel in the summation convention.

Role in Tensor Component Notation

Distinguishing Superscript from Subscript

A general tensor of type (p, q) is written with p superscript indices and q subscript indices, T^{i₁...iₚ}_{j₁...j_q}. The superscript indices label the contravariant slots — those that accept covectors (dual vectors) as arguments — while the subscript indices label the covariant slots, which accept ordinary vectors. The superscript index is not a power or exponent; its vertical position is purely notational and encodes transformation behavior, not repeated multiplication.

The Contravariant Transformation Law

Under a change of basis described by a Jacobian matrix, a tensor component carrying a superscript index transforms with the inverse Jacobian, in contrast to a subscript index, which transforms with the Jacobian itself. For a vector component v^i, the law reads

v^{i'} = \sum_{i} \frac{\partial x^{i'}}{\partial x^{i}} v^{i}

where x^{i} and x^{i′} denote the old and new coordinate functions. The presence of the partial derivative of the new coordinate with respect to the old — rather than the reverse — is the defining feature of contravariant, superscript-indexed transformation.

Superscript Indices and the Summation Convention

Pairing with a Subscript for Contraction

The Einstein summation convention stipulates that a superscript index repeated as a subscript index within the same term is summed over its full range, and the pair is said to be contracted. In T^{i}_{j} v^{j}, the superscript i remains free while the subscript j on T pairs with the superscript j on v, producing a sum over j and yielding an object with a single free superscript index i.

Why Only Superscript–Subscript Pairs Contract

A valid contraction always pairs one superscript with one subscript occurrence of the same letter; two superscripts or two subscripts sharing a letter do not form a legal implicit sum in this convention, because such a pairing would not be coordinate-invariant. The contravariant transformation factor attached to a superscript index is the matrix inverse of the covariant factor attached to a subscript index, so summing over a superscript–subscript pair causes the two Jacobian factors to cancel, leaving an expression that transforms correctly as a tensor. Summing two indices of the same type would leave an uncancelled Jacobian factor, breaking coordinate invariance.

\sum_{i} \frac{\partial x^{i'}}{\partial x^{i}} \frac{\partial x^{i}}{\partial x^{j'}} = δ_{j'}^{i'}

Free Versus Dummy Superscript Indices

Free Superscript Indices

A superscript index that appears exactly once in a term, with no matching subscript occurrence, is a free index: it is not summed, and it labels one component among the several components of the resulting tensor. The set of free superscript and subscript indices in an equation must match on both sides, term by term, for the equation to be a valid tensor equation.

Dummy Superscript Indices

A superscript index that is paired with a matching subscript index in the same term is a dummy (or bound) index: it is summed out and does not survive to label the result. Dummy indices may be renamed to any unused letter without changing the meaning of the expression, since the letter itself carries no information once the sum is performed; only the pattern of pairing matters.

Diagram of Index Placement

Superscript Indices in Common Tensor Objects

Vector Components

A vector v expressed in a coordinate basis has components v^{i} carrying a single superscript index, reflecting that a vector itself is the archetypal contravariant object: it is built from basis vectors e_{i} (subscript-indexed) via v = v^{i} e_{i}, where the superscript component index and the subscript basis index are contracted.

The Metric's Inverse

The inverse metric tensor g^{ij} carries two superscript indices and is used to raise indices, converting a subscript (covariant) index into a superscript (contravariant) one: v^{i} = g^{ij} v_{j}. This operation is the standard mechanism by which a covariant component is re-expressed as a contravariant one relative to a chosen metric.

Mixed Tensors

Tensors such as the Kronecker delta δ^{i}_{j} or the Riemann curvature tensor R^{i}_{jkl} combine superscript and subscript indices in a single object, with the superscript slots and subscript slots each transforming according to their own respective law; the total number of superscripts and subscripts fixes the tensor's type (p, q).

Consequences of Misplacing a Superscript Index

Loss of Tensorial Character

Writing an index as a subscript when it should be a superscript, or vice versa, changes the transformation law attached to that slot; an expression with a misplaced index generally fails to transform as a tensor under a change of coordinates, even if it is numerically equal to a correct tensorial expression in one particular coordinate system.

Detecting Errors via Balance

Because free indices must match in type and letter on both sides of a valid tensor equation, checking that every superscript on the left appears as a superscript on the right (and likewise for subscripts) is a standard and effective way to catch notational errors before any numerical computation is performed.