9.9.2 Tensor Coordinate Axis Assignment

Tensor Coordinate Axis Assignment is the decision fixing which independent coordinate direction of a space is designated as the first axis, which as the second, and so on through however many independent directions the space admits, establishing the correspondence between axis number and coordinate function before any basis vectors, dual bases, or tensor components can be built on top of that correspondence; it is the initial act of organization that the entire tensor coordinate system structure depends upon, settled before coordinate lines, basis vectors, or component arrays can be defined in relation to one another.

What Axis Assignment Fixes

A Numbering of Independent Directions

Axis assignment associates each coordinate function x^1, …, x^n with a numeral, fixing once and for all which independent direction of variation in the space is to be called "axis one," which "axis two," and so forth, with no two coordinate functions sharing the same assigned number.

A Prerequisite for Every Later Structural Element

Because coordinate lines, coordinate basis vectors, and component arrays are all defined relative to a specific coordinate function x^i, none of these later elements of the tensor coordinate system structure can be described without an axis assignment already having fixed which coordinate function is being referred to by the index i.

How Axis Assignment Is Typically Made

Following an Existing Convention

In common settings, axis assignment follows an established convention, such as assigning the first axis to horizontal displacement, the second to vertical displacement, and further axes to additional independent directions in the order they are introduced, so that the assignment matches expectations already familiar from prior use.

An Arbitrary but Fixed Choice in General Settings

In settings without an established convention, axis assignment may be made arbitrarily, with no particular direction more entitled than another to be called "the first axis"; what matters for the resulting tensor coordinate system structure is only that, once made, the assignment is fixed and applied consistently throughout.

Consequences of Changing an Axis Assignment

Component Arrays Are Permuted, Not Altered in Value

Reassigning which coordinate function is called axis one and which is called axis two, while keeping the underlying coordinate functions themselves unchanged, permutes the positions of entries within any component array built from that coordinate system, without changing which numerical value is attached to which actual coordinate direction.

Consistency Must Be Maintained Across an Entire Calculation

Any calculation that mixes tensors expressed using two different axis assignments produces meaningless results unless the assignments are first reconciled; axis assignment must therefore be settled once, at the start of a calculation, and held fixed for every tensor entering that calculation.

Diagram of Axis Assignment

Consequences of Axis Assignment for Tensor Coordinates

It Determines the Ordering Convention for Every Index That Follows

Because tensor component arrays, multi index addresses, and ordered tuple positions are all defined in terms of which coordinate function is axis i, the axis assignment made at the outset propagates through every subsequent index notation used in the tensor coordinate system, determining their ordering implicitly.

It Must Precede Every Other Element of the Coordinate System Structure

No other part of the tensor coordinate system structure — the induced basis, the coordinate grid, or the component arrays built from them — can be specified without an axis assignment already in place, making it the necessary first step in establishing the full structure rather than one component among equals.