12.1.2 Tensor Evaluation Operation Scope

Tensor Evaluation Operation Scope defines how tensors are computed within defined algebraic contexts, guiding expression interpretation and calculation.

Tensor Evaluation Operation Scope is the set of prerequisite conditions governing when a tensor, viewed as a multilinear map, can be evaluated on a specific collection of vector and covector arguments, requiring each argument to be plugged into a slot of the opposite variance type from the argument itself, with the number and type of arguments matching exactly the tensor's own variance type.

Foundational Setting

A Tensor as a Multilinear Function

Beyond being an indexed array of components, a tensor of type $(p, q)$ can be understood as a multilinear function that accepts $p$ covector arguments and $q$ vector arguments, producing a single scalar output. The evaluation operation is the act of feeding specific arguments into this function.

Why Scope Must Be Specified Here Too

Just as algebraic operations such as contraction have defined prerequisites, evaluation has its own scope: not every combination of vector and covector arguments can be fed into a given tensor, and the evaluation operation is defined only when the arguments match the tensor's variance type in both number and kind.

The Matching Requirement

Covectors Fill Contravariant Slots

Each of the $p$ upper, contravariant slots of a tensor must be filled by a covector argument, since a contravariant slot is, by construction, a linear functional-accepting position:

T (ω, \dots) = \sum_{i} T^{i} ω_{i}

Vectors Fill Covariant Slots

Correspondingly, each of the $q$ lower, covariant slots must be filled by a vector argument:

T (\dots, v) = \sum_{j} T_{j} v^{j}

Exact Count Requirement

The evaluation operation is undefined unless exactly $p$ covector arguments and exactly $q$ vector arguments are supplied, since supplying too few leaves some slots unfilled and produces a tensor rather than a scalar, while supplying too many has no defined slot to receive the extra argument.

Why the Matching Ensures a Well-Defined Result

Reversed Roles Would Break Invariance

Feeding a vector into a contravariant slot, or a covector into a covariant slot, is outside the operation's scope because doing so would not produce the cancellation of transformation factors that guarantees a basis-independent scalar result. Only the correct pairing, contravariant slot with covector argument and covariant slot with vector argument, yields the necessary cancellation:

\sum_{i} {\tilde{T}}^{i} {\tilde{ω}}_{i} = \sum_{i} T^{i} ω_{i}

Visual Overview

Diagram of Slot-Argument Matching

Partial Evaluation Within Scope

Filling Some but Not All Slots

The evaluation operation's scope also includes the case of filling only some of a tensor's slots, provided the arguments supplied still match the variance type of the specific slots they are assigned to, producing not a scalar but a tensor of reduced type corresponding to the remaining, unfilled slots:

S (ω) = \sum_{i} T_{j}^{i} ω_{i}

Consistency with Contraction

This partial evaluation is, in effect, the same operation as contraction viewed from the perspective of supplying an explicit argument rather than summing against another tensor's matching index, illustrating that evaluation and contraction share the same underlying scope requirement of pairing opposite variance types.

Summary of Key Traits

Defining Characteristics

Evaluation requires exactly as many covector arguments as a tensor has contravariant slots, and exactly as many vector arguments as it has covariant slots.
Contravariant slots must be filled by covectors, and covariant slots must be filled by vectors, never the reverse.
This matching is what guarantees the cancellation of transformation factors needed for a basis-independent result.
Partial evaluation, filling only some slots, remains within scope provided each supplied argument matches the variance type of its assigned slot, and reduces to the same logic underlying contraction.