9.19.1 Tensor Coordinate Input Selection

Tensor Coordinate Input Selection is the step within the coordinate calculation procedure at which the specific basis vectors and dual basis covectors to be substituted into the tensor for a given component are chosen, according to the index values currently being computed. It concerns picking out exactly which elements of the basis and dual basis serve as the inputs to the tensor at each stage of the calculation.

What Is Being Selected

Matching Inputs to Index Values

For each component being computed, input selection identifies the specific dual basis covector to supply for every contravariant index and the specific basis vector to supply for every covariant index, based directly on the numerical index values assigned to that component.

One Selection per Component

Input selection is performed anew for every component in the array being computed, since each component is associated with its own distinct combination of index values and therefore its own distinct combination of basis vectors and dual basis covectors.

Systematic Selection Across the Full Array

Enumerating All Required Combinations

To compute the entire component array of a tensor of type (p, q) over an n-dimensional space, input selection must be carried out for every one of the n raised to the power of p plus q possible combinations of index values, ensuring no valid combination is omitted.

Consistent Ordering of Selected Inputs

The order in which selected basis vectors and dual basis covectors are supplied to the tensor must match the order of the corresponding indices in the tensor's type, since supplying them out of order generally produces a component belonging to a different, unintended index pattern.

Correctness of the Selection

Selection Must Draw from a Single Coordinated Basis

All selected basis vectors and dual basis covectors for a given calculation must be drawn from one coordinated basis and its associated dual basis, since selecting from mismatched or inconsistent bases undermines the Kronecker pairing that the calculation relies upon.

Avoiding Repetition or Omission

Correct input selection avoids both repeating the same combination of basis elements for two different components and omitting a required combination altogether, since either error leaves the resulting component array incomplete or internally inconsistent.

Selection in Special Circumstances

Selection Under Known Symmetry

When a tensor's symmetry properties are already known, input selection can be restricted to only the independent combinations of basis vectors and dual basis covectors, since components related by the known symmetry can be filled in afterward without a separate selection and evaluation.

Selection During Basis Change

When recomputing components after a change of basis, input selection is instead applied to the new basis vectors and new dual basis covectors, with the old components and the transformation matrix taking the place of a direct reevaluation of the tensor against the new inputs.

Practical Importance of Input Selection

Foundation for Correct Component Values

Correct input selection is a prerequisite for the calculation procedure to yield the intended component array, since evaluating the tensor against the wrong combination of basis elements produces a value that does not belong in the slot it was meant to fill.

Supporting Systematic, Error-Free Computation

Approaching input selection systematically, by enumerating index combinations in a fixed, predictable order, helps ensure that a complete and correctly labeled component array is produced without gaps or misassigned entries.