7.13.3 Tensor Component Evaluation Method

Evaluating tensor components involves systematic methods to extract and interpret their values within a given coordinate system.

Tensor Component Evaluation Method is the actual computational procedure by which a tensor, once supplied with the basis vectors and covectors selected for a chosen index tuple, is carried out as a function to produce the resulting scalar value of that component.

The Nature of the Evaluation Step

From Selected Arguments to a Scalar Output

Evaluation is the step that follows index selection: having already chosen which basis elements will serve as arguments, evaluation is the concrete act of applying the tensor's defining rule to those arguments and computing the single number that results.

evaluation: (e_{i}, e_{j}) \to T (e_{i}, e_{j}) = T_{i j}

Dependence on How the Tensor Is Originally Defined

The specific method used for evaluation depends entirely on how the tensor was originally given, whether as an explicit formula, as a composition of simpler tensors, or as a geometric or physical quantity defined by some other rule, since each of these origins dictates a different concrete calculation to carry out.

Evaluation Methods by Tensor Origin

Direct Formula Evaluation

When a tensor is defined by an explicit algebraic formula involving its arguments, evaluation proceeds by substituting the chosen basis elements directly into that formula and simplifying the resulting expression down to a single number.

Evaluation via Known Building Blocks

When a tensor is built from simpler tensors through operations such as sums, products, or contractions, evaluation proceeds by evaluating each building-block tensor first and then combining those intermediate results according to the rule that defines the composite tensor.

(S \otimes T) (e_{i}, e_{j}) = S (e_{i}) \cdot T (e_{j})

Evaluation via Geometric or Physical Definition

When a tensor arises from a geometric or physical construction, such as a metric tensor defined from distances or a stress tensor defined from measured forces, evaluation instead relies on the specific rule connecting the underlying geometric or physical quantity to the numerical value assigned to each pair of basis directions.

Numerical Considerations in Evaluation

Exactness Versus Approximation

Depending on the source of the tensor, evaluation may yield an exact numerical value, as with tensors built from algebraic formulas involving whole numbers or simple fractions, or it may yield only an approximate value, as with tensors derived from empirical measurement or numerical simulation.

Consistency Across Repeated Evaluation

A properly defined tensor must yield the same evaluated component every time the same index tuple is evaluated relative to the same fixed basis, and any variation between repeated evaluations at the identical tuple signals either a computational error or an inconsistency in how the tensor has been defined.

Diagrammatic Illustration

The evaluation method pictured as a computational process taking selected basis arguments as input and producing a single scalar output.

Broader Significance of the Evaluation Method

Completing the Extraction Operation

Evaluation is the culminating step of the extraction operation, transforming a chosen index tuple from an abstract selection into a concrete numerical entry ready to be recorded in the component table, and without a well-defined evaluation method the extraction operation would remain incomplete.

Transferability Across Different Bases

Although the specific numbers produced by evaluation depend on the chosen basis, the underlying evaluation method itself, whether formula-based, compositional, or geometric, remains applicable unchanged when a different basis is selected, since only the identity of the supplied basis elements changes, not the rule governing how the tensor processes them.