12.2.4 Tensor Evaluation Operation Area

The Tensor Evaluation Operation Area defines how tensors are assessed within algebraic structures, establishing their functional role in mathematical transformations.

Tensor Evaluation Operation Area is the subset of tensor algebra concerned with treating a tensor as a multilinear function and supplying it with explicit vector and covector arguments to produce a scalar or a reduced-rank tensor, emphasizing the functional interpretation of tensors as maps acting on inputs rather than the purely component-level combination emphasized by the other operation areas.

Foundational Setting

A Tensor as a Function of Arguments

Beyond being an array of components indexed by upper and lower slots, a tensor of type $(p, q)$ can be regarded as a multilinear function accepting $p$ covector arguments and $q$ vector arguments, linear separately in each one, and returning a single number. Evaluation is the act of supplying such arguments and reading off the resulting value.

Distinguishing Evaluation from Additive and Composition Areas

Where the additive area combines tensors of matching type and the composition area joins tensors through a product-and-contraction pattern, evaluation is unique in requiring inputs, vectors and covectors, that are not themselves necessarily tensors of the same type as the object being evaluated, shifting the emphasis from tensor-to-tensor combination to tensor-as-function application.

Full Evaluation

Supplying Every Required Argument

When exactly $p$ covectors and $q$ vectors are supplied, filling every slot of the tensor, evaluation produces a single scalar:

T (ω, v) = \sum_{i, j} T_{j}^{i} ω_{i} v^{j}

Bilinear and Multilinear Forms as Evaluation Examples

A bilinear form acting on two vectors, and more generally a multilinear form acting on several vectors or covectors at once, is precisely this full evaluation operation applied to a tensor of the appropriate type.

Partial Evaluation

Filling Some but Not All Slots

Evaluation need not fill every slot at once: supplying fewer arguments than the tensor's full complement of slots produces not a scalar but a tensor of correspondingly reduced type, retaining the unfilled slots as free indices:

S (ω) = \sum_{i} T_{j}^{i} ω_{i}

Currying and Sequential Argument Supply

Because of multilinearity, supplying arguments to a tensor's slots one at a time, in any order, and combining the results is equivalent to supplying all arguments simultaneously, a property closely paralleling the notion of currying a function of several variables in general mathematics.

Visual Overview

Diagram of the Evaluation Process

Relationship to Contraction

The Same Operation, Two Perspectives

Full or partial evaluation is, at the level of the underlying computation, identical to contracting the tensor's relevant indices against the components of the supplied vector or covector arguments, so evaluation and contraction describe the same mathematical content viewed from two complementary perspectives, one emphasizing an external argument being supplied and the other emphasizing an internal summation between two tensors.

Why Both Perspectives Are Useful

The evaluation perspective is often more natural when a tensor is being used as a rule or a measuring device applied to specific inputs, such as a bilinear form measuring a pair of vectors, while the contraction perspective is more natural when combining two tensors that are both already part of a larger algebraic expression.

Summary of Key Traits

Defining Characteristics

Evaluation treats a tensor as a multilinear function, supplying covector arguments to its contravariant slots and vector arguments to its covariant slots.
Full evaluation, filling every slot, produces a scalar; partial evaluation produces a tensor of correspondingly reduced type.
Multilinearity allows arguments to be supplied sequentially or simultaneously with equivalent results, mirroring currying of ordinary multivariable functions.
Evaluation and contraction describe the same underlying computation from two complementary perspectives, external argument application versus internal tensor-to-tensor summation.