9.10.1 Tensor Coordinate Tuple Entry Order

Tensor Coordinate Tuple Entry Order is the fixed sequence in which the individual coordinate values of a point are listed within its coordinate tuple, matching the numbering already established by axis assignment, so that the first entry of a tuple always corresponds to the first coordinate axis, the second entry to the second axis, and so forth, without needing to be restated each time a tuple is written down; it is what allows a bare list of numbers such as (3, -1, 5) to be read unambiguously as referring to specific, individually identified coordinate directions.

What Entry Order Fixes

A Position-to-Axis Correspondence

Entry order establishes that the i-th position within a coordinate tuple always holds the value of the i-th coordinate function, so that a tuple (x^1(p), x^2(p), …, x^n(p)) can be written and interpreted without ambiguity about which entry belongs to which axis.

Consistency With Axis Assignment

Entry order is not an independent choice but is inherited directly from the axis assignment already fixed for the coordinate system; the tuple's entry order does no more than transcribe, into a written sequence, the same numbering that axis assignment previously attached to each coordinate direction.

Why a Fixed Entry Order Is Necessary

Numbers Alone Carry No Identity

A bare numeral appearing in a tuple carries no information, by itself, about which coordinate direction it measures; only its position within the tuple, combined with the fixed entry order convention, identifies which axis that numeral refers to.

Comparing Tuples Requires a Shared Order

Two coordinate tuples can be compared entry by entry, added, or otherwise combined only if both were written using the same entry order; comparing tuples produced under different entry order conventions without first reconciling them would compare unrelated quantities as if they corresponded.

Entry Order and Tensor Components

Vector Components Follow the Same Order as Position Tuples

Because a vector's components in a coordinate basis are assigned relative to the same coordinate basis vectors that the coordinate tuple's entry order was built to match, a vector's component array follows the identical entry order as a position tuple in that coordinate system, allowing the two to be handled with the same conventions.

Multi-Index Tuples Extend the Same Convention Per Factor

For higher-order tensors, the multi index address extends the same entry order convention independently to each factor of a tensor product, so that within each factor's own portion of the address, the coordinate tuple entry order convention continues to apply exactly as it does for a single coordinate tuple.

Diagram of Coordinate Tuple Entry Order

Consequences of a Fixed Entry Order

It Allows Tuples to Be Read Without Additional Labels

Because entry order is fixed in advance, a coordinate tuple can be written as a bare sequence of numbers, without repeating a label for every entry, while remaining fully interpretable to anyone already aware of the coordinate system's axis assignment and entry order convention.

Any Deviation Must Be Announced Explicitly

If a tuple is ever written using an entry order different from the established convention — for instance, listing coordinates in reverse — this deviation must be stated explicitly, since silently changing the entry order while continuing to use the same notation would cause the tuple to be misread according to the standard convention.