9.19.3 Tensor Coordinate Component Computation

Tensor Coordinate Component Computation is the central stage of the coordinate calculation procedure in which the tensor is actually evaluated against the selected basis vectors and dual basis covectors, producing the specific numerical or symbolic value assigned to each component. It is the step where the abstract tensor and the prepared basis are brought together to generate concrete output values.

The Computation Itself

Direct Evaluation Against Selected Inputs

For each component, computation consists of applying the tensor, as a multilinear map, to the specific combination of dual basis covectors and basis vectors identified during input selection, and recording the resulting scalar as the value of that component.

Computation From Prior Definitions

When the tensor is originally given through a formula, a bilinear form, or another underlying rule, computation substitutes the selected basis elements directly into that rule, carrying out whatever arithmetic or symbolic simplification the rule requires to produce the final scalar.

Computing an Entire Array

Repetition Across All Components

Full computation of a tensor's component array requires performing this evaluation once for every combination of index values permitted by the tensor's type, repeating the same underlying process with different selected inputs each time.

Accumulating Results Into the Array

As each individual value is computed, it is placed into its corresponding slot of the component array, indexed according to the same index pattern used to select its inputs, gradually building up the complete representation of the tensor in the chosen basis.

Computation via Transformation Instead of Direct Evaluation

Using Known Components and a Transformation Matrix

When components are already known in one basis and are needed in another, computation proceeds instead by applying the transformation law directly to the known components, using the transformation matrix or its inverse rather than reevaluating the tensor's original definition.

Equivalence of the Two Computation Routes

Whether computation proceeds by direct evaluation against a freshly prepared basis or by transforming previously known components, the resulting component array for a given basis is the same, since both routes ultimately compute the coefficients of the same underlying tensor relative to that basis.

Reducing Computational Effort

Exploiting Known Symmetry

Where a tensor is known in advance to be symmetric or antisymmetric in certain indices, computation can be limited to the independent components, with the remaining components filled in afterward according to the known symmetry relation rather than being separately evaluated.

Recognizing Zero Contributions Early

If part of the tensor's defining rule is known to vanish for a given combination of selected inputs, computation for that component can often be shortened or skipped entirely once this vanishing is recognized, rather than carrying out the full evaluation only to arrive at zero.

Practical Role of Component Computation

The Bridge to Usable Numerical Data

Component computation is what actually converts an abstract tensor, together with a prepared basis, into a concrete numerical array that can be used directly in further calculations, comparisons, or reporting.

Dependent on Correct Preparation and Selection

The correctness of every computed component depends entirely on the basis having been properly prepared beforehand and on the inputs having been correctly selected for that component, since computation itself simply carries out the evaluation using whatever inputs it is given.