9.2.2 Tensor Coordinate System Area

Tensor Coordinate System Area is the classification of coordinate systems along the axis of global structural reach and internal grid geometry — single-chart (global) systems that cover an entire space with one coordinate assignment versus multi-chart (atlas-based) systems required when no single assignment can, and orthogonal coordinate grids whose coordinate curves cross everywhere at right angles versus non-orthogonal grids that do not — complementing the classification of coordinate areas by the behavior of their basis vectors with a classification centered on the coordinate grid's own global and geometric organization. It addresses how many coordinate patches a space needs and how the resulting grid lines relate to one another, rather than how the basis vectors built from those coordinates behave under differentiation.

Single-Chart Versus Multi-Chart Coordinate System Areas

Spaces Coverable by One Coordinate Assignment

Some spaces admit a single coordinate system whose scope is the entire space at once; Euclidean space with standard Cartesian coordinates is the paradigm example, requiring no atlas at all since one coordinate assignment, injective and smoothly invertible everywhere, suffices for the whole space. Tensor calculations on such a space can proceed entirely within a single coordinate system area without ever needing a transition function.

Spaces Requiring an Atlas of Multiple Charts

Other spaces, most notably those with nontrivial topology such as spheres, admit no single coordinate system covering the whole space without some form of degeneracy; such spaces belong to the multi-chart coordinate system area, requiring an atlas of two or more overlapping coordinate assignments, with tensor components defined patch by patch and related across overlaps by the coordinate change transformation.

with each Uₐ a chart domain and no single Uₐ equal to all of M.

Orthogonal Versus Non-Orthogonal Coordinate Grids

Orthogonal Systems Simplify the Metric's Off-Diagonal Terms

An orthogonal coordinate system area is one in which the coordinate curves — the curves traced out by varying one coordinate while holding the others fixed — intersect at right angles everywhere; this property forces the metric tensor's off-diagonal components to vanish identically, leaving only diagonal entries to compute, which is the primary reason orthogonal systems such as polar, cylindrical, and spherical coordinates are favored whenever a problem's symmetry permits their use.

Non-Orthogonal Grids and Their Extra Bookkeeping

A non-orthogonal coordinate system area retains nonzero off-diagonal metric components, reflecting coordinate curves that cross at oblique angles; oblique (skewed affine) coordinates and many coordinate systems adapted to a specific non-rectangular physical structure fall into this area, requiring the full metric matrix, including its off-diagonal entries, to be tracked throughout any computation of lengths, angles, or index raising and lowering.

Diagram Comparing System Areas Along Both Axes

Interaction Between the Two Classification Axes

Independence of Chart Count and Grid Orthogonality

Whether a coordinate system requires a single chart or an atlas is a question about the global topology of the underlying space, while whether its grid is orthogonal is a question about the local geometric alignment of the coordinate curves; the two properties vary independently, so any combination — a single orthogonal chart, a single non-orthogonal chart, a multi-chart orthogonal atlas, or a multi-chart non-orthogonal atlas — is possible depending on both the space's topology and the specific coordinate functions chosen.

Choosing a System Area Based on the Task

Because orthogonal systems simplify metric computations while single-chart systems avoid the added bookkeeping of transition functions, the most convenient coordinate system area for a given task is typically the one nearest the single-chart, orthogonal corner of this classification that the underlying space and problem's symmetry actually permit, with departures toward non-orthogonal or multi-chart systems accepted only when the space's topology or the problem's structure genuinely requires them.