6.20.2 Tensor Covector Single Index Form

Tensor Covector Single Index Form is the notational convention of writing the components of a covector with exactly one subscript, φ_i, reflecting the type (0,1) classification and standing in explicit contrast to the single superscript form v^i used for vectors, a distinction that signals precisely how each object transforms under a change of basis. This single index form is the covariant counterpart to the vector's single index form, and recognizing the meaning carried by subscript placement, rather than treating it as an arbitrary typographic choice, is essential to reading tensor expressions correctly.

Reading the Single Index Form

Subscript Position and Its Meaning

The notation φ_i places the index i as a subscript, signaling unambiguously that φ is a type (0,1) covariant tensor; the position of the index, not merely its presence, carries this meaning, since the alternative form φ^i, with the index as a superscript, would instead denote a vector, a type (1,0) object governed by the opposite transformation law.

The Index as a Placeholder for a Range of Values

Written alone, φ_i represents the entire collection of components of φ, with i ranging implicitly over 1 through n; this single symbolic expression stands for n distinct numbers, φ_1, φ_2, ..., φ_n, once a specific dual basis has been chosen.

Expanding the Single Index Form into a Full Covector

Recovering the Abstract Covector from Its Components

The single index form is connected to the abstract, basis-independent covector φ through the expansion:

with the repeated index i, appearing once as a subscript on the components and once as a superscript on the dual basis covectors, summed according to the Einstein convention. This expansion justifies calling φ_i the components "of" φ: they are precisely the coefficients needed to reconstruct φ as a linear combination of the dual basis covectors e^i.

Row Vector Presentation

In numerical or computational contexts, the single index form φ_i is commonly displayed as a row of numbers, with the index i running across the row from 1 to n; this presentation is chosen so that the natural pairing φ(v) = φ_i v^i matches the convention of ordinary matrix multiplication, a row vector multiplying a column vector to produce a scalar.

Behavior of the Single Index Form Under Substitution

Contraction with an Upper Index

When the single index form of a covector appears alongside an object carrying a matching upper index, such as φ_i v^i or φ_i T^i_k, the repeated index i is summed, and the single index form disappears from the final expression, leaving either a scalar, in the case of φ_i v^i, or a new single index form with a different free index, in the case of φ_i T^i_k, which produces ψ_k.

Renaming Dummy Indices Freely

Because the specific letter used for the index in the single index form is arbitrary, φ_i and φ_j denote exactly the same covector, and dummy indices in a summed expression can always be renamed to any unused letter without changing its meaning, a manipulation frequently used to avoid clashes when combining several tensor expressions.

Diagram of the Single Index Form

Common Pitfalls Avoided by Careful Use of the Form

Confusing Subscript with Sequence Labeling

A frequent source of confusion is mistaking the subscript i in φ_i for an arbitrary labeling device, as used when numbering a sequence of unrelated quantities φ_1, φ_2, φ_3; in tensor notation, the subscript specifically indicates covariant transformation behavior, a meaning that must be understood from context and is distinct from mere sequential labeling.

Mismatched Index Forms Across an Equation

An equation that equates a single index form on one side with a superscripted expression on the other, without an appropriate metric to convert between them, signals an invalid tensor equation; correct tensor equations always match the variance of free indices exactly on both sides, meaning a subscript i on the left must correspond to a subscript i on the right, not a superscript.