9.11.1 Tensor Coordinate Basis Selection

Tensor Coordinate Basis Selection is the deliberate act of choosing, from among all coordinate systems and bases available for a given space, the particular one to be used as the starting point of the tensor coordinate representation process, guided by practical considerations such as the symmetry of the problem, the simplicity of the resulting components, or compatibility with a domain already fixed by other constraints; it is the initial decision that determines every subsequent tuple, component array, and reconstructed object produced by the representation process that follows it.

Criteria Guiding Basis Selection

Matching the Symmetry of the Problem

A basis is often selected because its coordinate directions align naturally with a symmetry already present in the problem, such as choosing a basis whose coordinate lines follow the natural axes of rotation or reflection a physical or geometric situation already exhibits, since this alignment tends to simplify the resulting tensor components considerably.

Minimizing Component Support

Basis selection may also be guided by the desire to minimize a tensor's component support, choosing a basis in which the tensor of primary interest has as few nonzero entries as possible, reducing the bookkeeping required for later calculations involving that tensor.

Compatibility With a Preexisting Domain Restriction

When the region of interest is already constrained by some other consideration, such as being bounded by particular surfaces, basis selection favors a coordinate system whose coordinate surfaces match those boundaries, since components and equations tend to simplify substantially when expressed relative to a basis compatible with the boundary shape.

Selection Does Not Change the Underlying Object

The Object Selected For Remains the Same Regardless of the Basis

Basis selection determines only how a tensor will subsequently be represented, not what the tensor itself is; the same abstract tensor admits representation in any admissible basis, so selection is properly understood as a decision about convenience of representation rather than about the object being represented.

Selection Precedes but Does Not Predetermine the Outcome of Assignment

Choosing a basis fixes which coordinate system will be used for component assignment, but it does not itself determine what the resulting components will be; those values still depend on the actual tensor being expressed, computed only once basis selection has been completed and component assignment is carried out.

Consequences of a Poor Basis Selection

Unnecessarily Large Component Support

Selecting a basis unrelated to any symmetry or convenient feature of the problem often results in a tensor's components filling most or all of its component support unnecessarily, obscuring structural properties of the tensor that a better-chosen basis would have revealed directly.

Increased Complexity in Later Stages of the Representation Process

A poorly selected basis can propagate additional complexity into every subsequent stage of the coordinate representation process, since tuple representation, array construction, and reconstruction must all operate on components that could have been simpler had a more suitable basis been selected at the outset.

Diagram of Basis Selection

Consequences of Basis Selection for the Representation Process

It Sets the Terms for Every Later Stage of Representation

Because tuple representation, array construction, and reconstruction all operate relative to whatever basis was selected, the choice made during basis selection propagates through the entire coordinate representation process, making this initial decision consequential for the practicality of every stage that follows.

It Can Be Revisited by Applying the Transformation Context

If a basis selected initially proves inconvenient once the representation process is underway, the tensor coordinate basis transformation context provides the means to switch to a better-suited basis without repeating the entire process from the original tensor, converting existing components directly into the new, more convenient basis.