12.9.3 Tensor Evaluation Scalar Result Case

Tensor Evaluation Scalar Result Case is the situation in which every one of a tensor's argument slots is filled with an appropriate vector or covector, fully saturating the multilinear map and producing a single scalar as the final, unambiguous output of the evaluation.

Conditions for the Scalar Case

Full Saturation of All Slots

For a tensor $$A$$ of type $$(p,q)$$, the scalar result case occurs precisely when all $$p$$ covector slots and all $$q$$ vector slots have been supplied with appropriate arguments, leaving no open positions:

Since every index of the tensor has now been contracted against a corresponding vector or covector component, no free index remains, and the result carries no further tensorial structure.

Componentwise Reduction to a Single Number

Because every index has been summed against a supplied argument, the componentwise expression for evaluation collapses entirely into a single sum of products, yielding one number rather than an array of numbers:

Basis Independence of the Scalar

The Number Produced Does Not Depend on Coordinates

Although the componentwise computation of $$c$$ appears to rely on a chosen basis, the resulting scalar is identical no matter which basis was used, since the transformation laws governing the tensor's components and the components of the supplied vectors and covectors cancel exactly during the full contraction.

Contrast with the Tensor Result Case

This full basis independence distinguishes the scalar result case from the tensor result case that arises under partial evaluation, since a partially evaluated tensor still carries free indices whose components genuinely depend on the basis chosen, whereas a fully evaluated scalar carries no such dependence at all.

Interpretive Significance

Scalars as Meaningful Outputs

The scalar produced by full evaluation often represents a meaningful geometric or physical quantity, such as a length, an angle-related quantity, an energy, or another measurable value, depending on the specific tensor and the vectors or covectors supplied to it.

Relation to Familiar Constructions

Many familiar constructions, such as the inner product of two vectors or the action of a linear functional on a vector, are specific instances of the scalar result case, where a tensor of low rank is fully evaluated on exactly the arguments needed to produce a single number.

Illustration