9.6.5 Tensor Noncoordinate Basis Coordinate Boundary

Tensor Noncoordinate Basis Coordinate Boundary is the precise dividing line separating frames that qualify as noncoordinate from frames that, despite perhaps not looking like an obvious coordinate basis at first glance, actually do arise from some coordinate system in a neighborhood of a point; it identifies the exact condition — vanishing of all structure functions throughout a neighborhood — that must hold for a frame to cross from the noncoordinate side of this boundary back to the coordinate side.

The Condition Defining the Boundary

Structure Functions Must Vanish Identically

A frame lies on the coordinate side of the boundary within some neighborhood exactly when its structure functions vanish identically throughout that neighborhood, so that every pair of frame members commutes everywhere in the region under consideration, not merely at an isolated point.

Vanishing at a Single Point Is Not Sufficient

The coordinate boundary is a condition on a whole neighborhood, not on an isolated point: a frame whose structure functions happen to vanish at one particular point, while remaining nonzero nearby, still lies on the noncoordinate side of the boundary, since no coordinate functions can be found whose derivatives reproduce the frame throughout any neighborhood of that point.

Why Vanishing Brackets Are the Correct Test

Coordinate Vector Fields Always Commute

Any set of vector fields obtained as partial derivatives with respect to a common set of coordinate functions automatically commutes, because mixed partial derivatives of a coordinate function are equal regardless of the order of differentiation; this is what makes vanishing brackets a necessary condition for lying on the coordinate side of the boundary.

Vanishing Brackets Are Also Sufficient

A local converse holds as well: whenever a frame's structure functions vanish throughout a neighborhood, coordinate functions can in fact be constructed, in a possibly smaller neighborhood, whose associated partial-derivative vector fields reproduce that frame exactly, so vanishing brackets are not just necessary but also sufficient to place a frame on the coordinate side.

Crossing the Boundary in Practice

A Frame Constructed for Convenience May Still Be Coordinate

A frame originally introduced without any coordinate functions in mind — chosen, for instance, for its algebraic convenience — may nonetheless turn out to lie on the coordinate side of the boundary if a direct check shows its structure functions vanish; the boundary depends only on the bracket condition, not on the manner in which the frame was originally introduced.

A Frame That Appears Coordinate-Like May Still Be Noncoordinate

Conversely, a frame that superficially resembles a coordinate basis, or that is built by simple algebraic combinations of a genuine coordinate basis, can still fail the vanishing-bracket condition and lie on the noncoordinate side, if those combinations reintroduce nonzero structure functions.

Diagram of the Coordinate Boundary

Consequences of the Coordinate Boundary

It Tells When Coordinate Simplifications Are Legitimately Available

Establishing that a given frame lies on the coordinate side of the boundary in some neighborhood licenses the use of coordinate functions and every simplification that follows from vanishing brackets throughout that neighborhood; establishing that it lies on the noncoordinate side rules out these simplifications and requires structure functions to be retained.

It Localizes Where a Frame's Character May Change

Because the boundary condition is checked neighborhood by neighborhood, it is possible for a single frame to lie on the coordinate side in one region of a space and on the noncoordinate side in another, so the coordinate boundary must be examined separately wherever the frame's structure functions are evaluated.