9.9.3 Tensor Coordinate Label Set

A tensor coordinate label set defines how indices identify components in tensor spaces, providing structure for algebraic operations and geometric interpretations.

Tensor Coordinate Label Set is the underlying set of admissible index values available for use in a given tensor coordinate system, most commonly the set of integers running from one to the dimension of the space, over which any single tensor index is understood to range whenever it is written without further qualification; it is the raw pool of labels that axis assignment draws upon to name individual coordinate directions, and that summation, contraction, and component enumeration draw upon whenever an index is required to run through every admissible value.

What the Label Set Consists Of

A Finite Set Matching the Dimension

For a space of dimension n, the coordinate label set is the finite set of integers from 1 to n, with exactly as many labels available as there are independent coordinate directions in the space.

L = {1, 2, \dots, n}

No Inherent Order or Meaning Attached to the Labels Themselves

Considered purely as a set, the coordinate label set carries no information about which label corresponds to which coordinate direction; that correspondence is supplied separately by the axis assignment, while the label set itself supplies only the raw collection of values available to be assigned.

The Role of the Label Set in Index Notation

Fixing the Range of Summation

Whenever an index is summed over, as in forming a contraction or an expansion of a tensor in a basis, the coordinate label set specifies exactly which values the summation index runs through, ensuring that every relevant basis direction is included exactly once and no value outside the label set is mistakenly included.

\sum_{i \in L} T^{i} e_{i}

Fixing the Size of a Component Array

Because every free index of a tensor's component array ranges over the same coordinate label set, the total number of entries in a component array of a given order is determined directly by the size of the label set raised to a power equal to the number of free indices.

| L |^{k}

Distinguishing the Label Set From Related Notions

Label Set Versus Axis Assignment

The coordinate label set supplies only the pool of values {1, …, n}; axis assignment is the separate act of deciding which specific coordinate function is designated by each of these values. The two together, but not either alone, are needed to fully specify what an index such as i refers to.

Label Set Versus Multi Index Address

A multi index address, used in a tensor product basis, is built from tuples whose entries are each drawn from one or more coordinate label sets, one per factor; the label set is the raw material from which such addresses are constructed, rather than being the address itself.

Diagram of the Coordinate Label Set

Consequences of a Well Defined Label Set

It Guarantees Complete and Unambiguous Enumeration

Because every index in a tensor coordinate system draws its values from the same fixed coordinate label set, any procedure requiring every component of a tensor to be enumerated, such as verifying a tensor equation componentwise, can proceed with confidence that no admissible index value has been overlooked and no inadmissible value has been included.

Changing Dimension Requires a New Label Set

Should the dimension of the space under consideration change — for example, when passing to a subspace or an extended space — the coordinate label set must be redefined to match the new dimension, and every index range, summation, and component array size depending on the old label set must be reconsidered accordingly.