9.4.1 Tensor Product Basis Coordinate Grid

Tensor Product Basis Coordinate Grid is the lattice-like arrangement of coordinate lines, surfaces, or higher-dimensional level sets that results when a coordinate system on a product space is built directly from the coordinate systems of its factor spaces, with each grid line or grid surface corresponding to holding all but one factor's coordinates fixed while that one factor's coordinate varies. It is the geometric picture underlying the tensor product basis element set: just as that element set is indexed by combinations of index values drawn independently from each factor, the coordinate grid is the corresponding geometric picture of combinations of coordinate values drawn independently from each factor space.

Constructing the Grid From Factor Coordinates

Combining Independent Coordinate Systems

Given two spaces M and N with their own coordinate systems x¹, ..., xᵐ on M and y¹, ..., yⁿ on N, the product space M × N receives a natural coordinate system consisting of both sets of coordinates together, (x¹, ..., xᵐ, y¹, ..., yⁿ); the resulting coordinate grid is the pattern formed by the level sets of each individual coordinate function, crossing one another to tile the product space into cells.

Grid Cells as Tensor Product Basis Elements

At any point of the product space, the coordinate basis vectors from the two factors — ∂/∂xⁱ tangent to the M directions and ∂/∂yʲ tangent to the N directions — together span the tangent space of the product, and the induced tensor product basis elements ∂/∂xⁱ ⊗ ∂/∂yʲ correspond directly to the individual grid cells formed by the crossing coordinate lines, tying the algebraic basis element set directly to the geometric grid structure.

Grid Lines and Their Relationship to the Factor Spaces

Lines Along One Factor Hold the Other Fixed

A grid line corresponding to varying xⁱ alone, with every other coordinate (including all of the y coordinates) held fixed, sweeps out a curve that lies entirely within a single copy of M, embedded in the product space at the fixed value of the N coordinates; symmetrically, a grid line varying some yʲ alone lies entirely within a single copy of N. The coordinate grid is therefore built from two families of lines, each family associated with one factor and running orthogonally, in the coordinate sense, to the other family.

The Grid Reflects the Product Structure Directly

Because the grid lines separate cleanly into an M-family and an N-family with no mixing between them, the coordinate grid of a genuine tensor product coordinate system directly visualizes the product structure of the underlying space, in contrast to a general coordinate system on a non-product space (or a product space using non-adapted coordinates), whose grid lines need not separate so cleanly along any such factorization.

Diagram of a Tensor Product Coordinate Grid

Metric Behavior on the Product Grid

Block-Diagonal Metric When Factors Are Metrically Independent

When the product space carries a metric built independently from a metric on each factor — the standard product metric — the metric components mix M-directions and N-directions in no way at all, so that the metric matrix expressed in the product coordinate grid is block-diagonal, with one block coming entirely from M's own metric and the other entirely from N's; this block-diagonal structure is a direct reflection of the grid's clean separation into two non-mixing families of lines.

Consequences for Contraction and Index Raising

Because the metric does not mix indices from the two factors, raising or lowering an M-type index uses only M's own metric block, and likewise for N-type indices, so that tensor operations respecting the product coordinate grid's structure can generally be carried out separately within each factor and only combined afterward, mirroring the same separation seen in the grid lines and in the tensor product basis element set itself.

Practical Use of the Product Grid Structure

Simplifying Calculations That Respect the Product Structure

Recognizing that a coordinate grid comes from a genuine tensor product of factor coordinate systems allows a calculation to be split along the grid's natural separation, handling the M-directions and N-directions largely independently and combining results only at the point where a genuinely mixed (cross-factor) quantity is required, which is typically far simpler than treating the product space's coordinates as an undifferentiated whole.

Recognizing When the Grid Does Not Reflect a Clean Product

If a coordinate system on what is nonetheless a product space is chosen without respecting the factor structure — for instance, a rotated or otherwise mixed coordinate system — the resulting grid lines will not separate cleanly into the two non-mixing families described above, and the simplifications associated with a genuine tensor product basis coordinate grid, including the block-diagonal metric, will not be available until the coordinates are changed back to ones adapted to the product structure.