6.19.2 Tensor Vector Single Index Form

Tensor Vector Single Index Form is the notational convention of writing the components of a vector with exactly one superscript, v^i, reflecting the type (1,0) classification and standing in explicit contrast to the single subscript form φ_i used for covectors, a distinction that carries essential meaning about how each object transforms under a change of basis. This single index form is the simplest nontrivial index expression in tensor algebra, and mastering its correct use, expansion, and interpretation is the necessary first step before working with the more elaborate multi-index expressions that arise for higher-order tensors.

Reading the Single Index Form

Superscript Position and Its Meaning

The notation v^i places the index i as a superscript, signaling unambiguously that v is a type (1,0) contravariant tensor; the position of the index, not merely its presence, carries this meaning, since the alternative form v_i, with the index as a subscript, would instead denote a covector, a type (0,1) object governed by an entirely different transformation law.

The Index as a Placeholder for a Range of Values

Written alone, v^i represents the entire collection of components of v, with i ranging implicitly over 1 through n, where n is the dimension of the vector space; this is a single symbolic expression standing for n distinct numbers, v^1, v^2, ..., v^n, once a specific basis has been chosen.

Expanding the Single Index Form into a Full Vector

Recovering the Abstract Vector from Its Components

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

with the repeated index i, appearing once as a superscript on the components and once as a subscript on the basis vectors, summed according to the Einstein convention. This expansion is what justifies calling v^i the components "of" v: they are precisely the coefficients needed to reconstruct v as a linear combination of the basis vectors e_i.

Column Vector Presentation

In numerical or computational contexts, the single index form v^i is commonly displayed as a column of numbers, with the index i running down the column from 1 to n; this presentation is a matter of typographic convenience, chosen to align naturally with the convention that a type (1,1) operator's matrix acts on a vector by multiplying it on the right as a column, [T][v].

Behavior of the Single Index Form Under Substitution

Contraction with a Lower Index

When the single index form of a vector appears alongside an object carrying a matching lower index, such as φ_i v^i or T^k_i v^i, 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 T^k_i v^i, which produces w^k.

Renaming Dummy Indices Freely

Because the specific letter used for the index in the single index form is arbitrary, v^i and v^j denote exactly the same vector, and dummy indices appearing in a summed expression can always be renamed to any unused letter without changing the meaning of the expression, a manipulation frequently used to avoid clashes when combining several tensor expressions into one.

Diagram of the Single Index Form

Common Pitfalls Avoided by Careful Use of the Form

Confusing Superscript with Exponent

A frequent source of confusion when first encountering the single index form is mistaking the superscript i in v^i for an exponent, as in v raised to the power i; the tensor notation convention reserves superscripts for contravariant indices specifically, and this usage must be inferred from context, since the same typographic symbol serves both purposes in general mathematical writing.

Mismatched Index Forms Across an Equation

An equation that equates a single index form on one side with a subscripted 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 superscript i on the left must correspond to a superscript i on the right, not a subscript.