13.11.2 Tensor Vector Covector Summed Index

Tensor Vector Covector Summed Index is the single shared index appearing once as a contravariant superscript on a vector and once as a covariant subscript on a covector, whose repetition under the Einstein summation convention signals that the two objects are to be paired term by term and added together to produce their contraction. It denotes the specific index responsible for linking a vector and a covector into a scalar, distinguishing this one summed label from any other indices that might appear elsewhere in a larger expression.

Conceptual Basis

The Repeated Index as a Summation Instruction

In the Einstein summation convention, an index that appears exactly once as a superscript and once as a subscript within the same term is understood to be summed over its full range without an explicit summation symbol. The vector covector summed index is precisely such a repeated label, instructing that the corresponding components of the vector and covector be multiplied and accumulated.

Requirement of Opposite Variance

The summed index must appear as a contravariant index on the vector and as a covariant index on the covector, since this opposite variance is what allows the transformation factors introduced by a change of basis to cancel, yielding a basis-independent scalar as the final result.

A Single Index, Fully Consumed

Because both the vector and the covector are rank-one objects, the vector covector contraction involves exactly one summed index, and once the summation is carried out, no indices remain on either side, leaving a scalar with no further tensorial structure.

Formal Description

Notation and Range

For a vector $$v^i$$ and a covector $${\omega }_i$$ defined over an $$n$$-dimensional space, the summed index $$i$$ ranges over the values $$1,2,\dots ,n$$, and the contraction is written:

with the summation over $$i$$ implicit in the repeated appearance of the index.

Explicit Expansion

Written without the summation convention, the same expression becomes:

showing explicitly the individual terms contributed by each value taken by the summed index.

Invariance of the Label Itself

Because the summed index is a dummy variable, replacing it with any other symbol not already in use elsewhere in the expression leaves the value of the contraction unchanged, so $${\omega }_i{v}^i$$ and $${\omega }_k{v}^k$$ denote exactly the same scalar.

Properties

Dummy Index Status

The summed index carries no meaning outside the specific term in which it appears, and unlike a free index, it does not survive into the result or affect the type of the resulting object, distinguishing it clearly from any other indices that might be present in a larger surrounding expression.

Requirement of Non-Repetition Elsewhere

Within a single term, an index intended as a summed index must not simultaneously be used as a free index elsewhere in that same term, since doing so would create ambiguity about whether the index is meant to be summed or held fixed.

Uniqueness in the Elementary Case

For the basic vector covector contraction, exactly one summed index exists, but in more elaborate expressions involving several such contractions, multiple distinct summed indices may appear, each requiring its own distinct dummy label to avoid confusion.

Practical Considerations

Avoiding Notational Ambiguity

When several vector covector contractions appear within a single larger expression, care must be taken to assign a distinct dummy label to each summed index, since reusing the same label for two unrelated summations would incorrectly imply that they are linked.

Role in Verifying Well-Formed Expressions

Checking that every occurrence of the summed index appears exactly once as a superscript and once as a subscript within its term is a standard method of verifying that a tensor expression involving vector covector contraction has been written correctly according to the summation convention.

Foundation for More General Contracted Indices

The vector covector summed index represents the simplest case of the more general notion of a contracted index appearing in higher-rank tensor contractions, where the same requirement of opposite variance and single repetition governs any index selected for summation.