9.1.2 Tensor Coordinate Assignment Scope

Tensor Coordinate Assignment Scope is the precise domain over which a coordinate map — the function assigning an ordered tuple of real numbers to each point of a region of a manifold — is required to be a well-defined, injective, and smoothly invertible correspondence, since it is exactly this domain, and no larger set, over which the assignment can be trusted to identify points uniquely and to support differentiation of tensor fields expressed through it. It is the most concrete layer of coordinate scope: rather than asking broadly how far a basis or representation extends, it isolates the specific requirement that the coordinate-assigning function itself remain one-to-one and smoothly invertible throughout its stated domain.

What the Assignment Must Satisfy Within Its Scope

Injectivity Across the Assigned Domain

A coordinate assignment x: U → ℝⁿ, mapping an open region U of a manifold to n-tuples of real numbers, must be injective throughout U: no two distinct points of U may be assigned the identical coordinate tuple, since a shared coordinate tuple would make it impossible to distinguish the two points using that coordinate system at all. The assignment scope is, by definition, restricted to exactly the region over which this injectivity genuinely holds.

Smooth Invertibility Within the Scope

Beyond mere injectivity, a coordinate assignment used for tensor calculus must have a smooth inverse on its image, so that the resulting coordinate functions can be differentiated and combined into a coordinate basis of tangent vectors at every point of U; the assignment scope is therefore the region where both the forward assignment and its inverse are smooth, not merely the (possibly larger) region where the forward map alone happens to be defined.

Where Assignment Scope Falls Short of the Full Manifold

Global Injectivity Frequently Fails

A coordinate assignment that is injective and smoothly invertible on a small enough region often fails to remain injective if its domain is extended too far; the standard longitude coordinate on a circle, for instance, can be assigned injectively only up to (but not including) a full revolution, since continuing past that point necessarily reassigns a coordinate value already used. The assignment scope for such a coordinate is exactly the sub-region short of this repetition, not the whole circle.

Genuine Topological Limits Versus a Poor Choice of Map

Sometimes a coordinate assignment's limited scope reflects nothing more than an unfortunate choice of coordinate function, remediable by choosing a better one; but for some manifolds, no coordinate assignment whatsoever can have scope equal to the entire space while remaining injective and smoothly invertible, a genuine topological obstruction rather than a fixable shortcoming of any particular assignment — this is exactly the circumstance that necessitates an atlas of multiple, separately scoped coordinate assignments covering the manifold together.

Diagram of an Assignment's Scope Falling Short of Full Injectivity

Consequences for Tensor Components Built From the Assignment

Components Inherit the Assignment's Scope Exactly

Any coordinate basis, and any tensor component array expressed relative to it, is defined precisely on the coordinate assignment's scope and nowhere beyond it; a tensor component formula derived using one coordinate assignment therefore carries an implicit domain restriction to that assignment's scope, and applying the formula outside that scope — for instance, at or beyond the point where injectivity fails — produces components that either do not exist or no longer correspond to the intended point.

Assignment Scope as the Root Cause of Broader Scope Limitations

The more general limitations discussed under basis and coordinate scope — chart domains not covering a whole manifold, the need for an atlas, coordinate singularities — all trace back to this same underlying fact about coordinate assignment scope: whatever region a coordinate assignment can maintain injectivity and smooth invertibility over is the full and final extent of that assignment's usefulness for representing tensors, and every broader statement about scope in tensor bases and coordinates is ultimately a restatement or consequence of this single, concrete requirement.