9.17.3 Tensor Basis Independent Evaluation Result

Tensor Basis Independent Evaluation Result computes tensor properties without a specific basis, yielding intrinsic structural results.

Tensor Basis Independent Evaluation Result is the scalar outcome produced when a tensor is evaluated against a full set of vector and covector arguments, understood specifically as a value that does not depend on any basis used along the way to compute it. It names the numerical output of such an evaluation as a basis-independent fact, distinguishing it from the basis-dependent intermediate quantities that may have been used to arrive at it.

Nature of the Evaluation

A Tensor as a Function of Its Arguments

A tensor of type (p, q) acts as a multilinear function taking p covector arguments and q vector arguments and producing a single scalar. This scalar output is the evaluation result, and it is defined directly through the tensor's action on its arguments, without reference to any basis.

T (ω, v) = c

Independence from the Method of Computation

Whether the scalar c is computed by directly applying the tensor to its arguments as abstract objects, or by expanding the tensor and its arguments in some basis and carrying out the corresponding sum over components, the resulting value is the same, since both methods compute the same well-defined function applied to the same arguments.

Computing the Result Through Components

Reduction to a Sum Over Components

In practice, the evaluation result is most often computed by expanding the tensor and its arguments in a chosen basis and summing the products of matching components, yet the value obtained this way is guaranteed to match the value obtained through any other basis.

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

Cancellation of Basis Dependence

Although the individual components of the tensor and its arguments each depend on the chosen basis, the specific combination formed by this sum is arranged so that all basis-dependent transformation factors cancel exactly, leaving a result that is the same in every basis.

Recognizing a Genuine Basis Independent Result

Full Saturation of Indices

An evaluation result is basis independent precisely when every index of the tensor has been paired with a matching argument, upper indices with covector arguments and lower indices with vector arguments, leaving no free indices remaining in the final expression.

Partial Evaluations Remain Basis Dependent

If only some of a tensor's arguments are supplied, leaving some indices unsaturated, the outcome is not a single scalar but another tensor of lower order, whose components still depend on the basis in the ordinary way, and such a partial evaluation does not qualify as a basis independent evaluation result.

Significance of the Result

A Reliable Point of Comparison

Because the evaluation result does not depend on the basis used to compute it, it provides a fixed point of comparison across calculations that may have used entirely different bases, making it possible to check the consistency of independent computations against one another.

Anchoring Abstract Definitions to Concrete Numbers

The evaluation result is what ultimately connects the abstract, basis-independent definition of a tensor as a multilinear map to concrete, checkable numbers, since it is a genuine number that can be computed, compared, and verified regardless of which basis was used to obtain it.