3.18.2 Tensor Covector Index Notation

Tensor Covector Index Notation is the specific system of subscripted indices used to represent the components of a covector and to encode, through the placement and repetition of those indices, the rules governing linear pairing, contraction, and transformation under change of basis. A covector ω belonging to the dual space V* of a vector space V is written in components as ωᵢ, with the index i running from 1 to n = dim(V), and the full covector is recovered from its components via ω = ωᵢeⁱ, where eⁱ denotes the i-th element of the dual basis. The lower placement of the index is not a stylistic choice but a precise bookkeeping device distinguishing covariant quantities from the upper-indexed components of ordinary vectors.

The Grammar of Index Placement

Lower Indices Signal Covariance

An index written as a subscript, as in ωᵢ, indicates that the quantity transforms covariantly, meaning its components change under a basis transformation in the same direction as the basis vectors themselves. This is codified precisely: if the basis changes by ẽⱼ = Aᵏⱼeₖ, then covector components change by ω̃ⱼ = ωᵢAⁱⱼ, using the same matrix A, rather than its inverse, that defines the new basis vectors.

Upper Indices Signal Contravariance

By contrast, vector components vʲ carry a superscript and transform by the inverse matrix, ṽʲ = (A⁻¹)ʲₖvᵏ. The notational asymmetry between ωᵢ and vʲ is a direct visual encoding of this opposite, contragredient transformation behavior, allowing the correctness of an expression to be partially verified just from the pattern of raised and lowered indices, independent of any deeper calculation.

The Einstein Summation Rule in Index Notation

Repeated Index Contraction

When an index appears exactly twice in a single term, once raised and once lowered, it is understood to be summed over its entire range without an explicit summation symbol. The canonical example is the pairing of a covector with a vector,

The convention applies exclusively to one raised and one lowered occurrence of the same index; two lowered or two raised copies of an index appearing in a term, such as ωᵢvᵢ written with both subscripted, are not summed under this convention and, more importantly, would not correspond to a basis-independent quantity if they were.

Free Indices and Tensorial Equations

An index appearing only once in every term of an equation is a free index and must appear at the same height, upper or lower, and with the same name on both sides of the equation for the equation to be a legitimate tensorial statement. For example, the pullback component formula (T*ω)ⱼ = ωᵢTⁱⱼ has j as a free lower index on both sides and i as a contracted dummy index, confirming the formula is well formed.

Index Notation for the Dual Basis

Biorthogonality

The dual basis covectors eⁱ satisfy eⁱ(eⱼ) = δⁱⱼ, where the Kronecker delta δⁱⱼ itself is written with one upper and one lower index, reflecting that it represents the identity map on V expressed as a mixed (1,1) tensor. This single relation is the generative rule from which the entire index notation for covectors is derived: it fixes how the components ωᵢ of an arbitrary covector are recovered by direct evaluation, ωᵢ = ω(eᵢ).

Change of Basis for the Dual Basis Itself

Under a change of basis on V by matrix A, the dual basis transforms contragrediently by the inverse transpose, ẽⁱ = (A⁻¹)ⁱⱼeʲ, which is precisely what is required to preserve the biorthogonality relation ẽⁱ(ẽⱼ) = δⁱⱼ in the new basis. This transformation of the dual basis vectors is the structural counterpart to, and opposite in form from, the transformation of the covector components ωᵢ themselves.

Index Notation in Pullback and Composition

Pullback in Components

For a linear map T: V → W with matrix components Tⁱⱼ, meaning T(eⱼ) = Tⁱⱼfᵢ relative to bases {eⱼ} of V and {fᵢ} of W, and a covector ω on W with components ωᵢ, the pullback T*ω has components

The index i, belonging to W, is contracted away, and only the index j, belonging to V, survives as free, correctly reflecting that T*ω is a covector on V, not on W.

Composition of Pullbacks in Index Notation

If S: U → V and T: V → W are linear maps with components Sᵏⱼ and Tⁱⱼ respectively, the components of the pullback along the composite T∘S are obtained by chaining the contraction, ((T∘S)ω)ₖ = ωᵢTⁱⱼSʲₖ, with the intermediate index j contracted between the two matrices in exactly the order matching how S and T are applied to vectors, providing an index-level confirmation of the identity (T∘S) = S∘T.

Index Notation for Higher-Rank Covariant Tensors

Multiple Lower Indices and Symmetry Properties

A covariant tensor of rank k is written with k lower indices, Tᵢ₁...ᵢₖ, and its symmetry type is read directly from how the notation behaves under index permutation: Tᵢⱼ = Tⱼᵢ marks a symmetric tensor, such as a metric, while Tᵢⱼ = −Tⱼᵢ marks an antisymmetric tensor, such as a differential two-form. Contraction of any such tensor with a corresponding number of vectors proceeds by summing each lower index against a matching upper index from a distinct vector, generalizing the single-index pairing rule that defines covector index notation in the rank-one case.