10.18.4 Tensorial Rule Coordinate Change Compatibility

The Tensorial Rule ensures compatibility under coordinate changes, preserving tensor properties across different reference frames.

Tensorial Rule Coordinate Change Compatibility is the property that the tensorial transformation rule applies consistently and without contradiction to any sequence or combination of admissible coordinate changes, so that transforming a tensor through several intermediate charts always agrees with transforming it directly between the initial and final charts, making the rule fit together with itself under composition, inversion, and identity in a self-consistent way.

Compatibility With Composed Transformations

Agreement Through an Intermediate Chart

If a tensor is transformed from a source chart to an intermediate chart, and then from the intermediate chart to a final target chart, the tensorial rule guarantees this two-step result matches the single direct transformation between the source and target charts:

J_{i}^{k} = \sum_{m}^{n} {\overset{'}{J}}_{i}^{m} {\overset{''}{J}}_{m}^{k}

with the primed Jacobians denoting the two intermediate steps; this is a direct consequence of the multivariable chain rule applied to the composition of the two transition maps, and it guarantees that no ambiguity arises from choosing to route a transformation through an intermediate chart rather than performing it in a single step.

Extension to Any Number of Intermediate Charts

The same compatibility extends inductively to a chain of any finite number of intermediate charts, with the composed Jacobian at each stage built from the product of the individual stage Jacobians, so the tensorial rule remains internally consistent regardless of how many intermediate coordinate systems a computation happens to pass through.

Compatibility With Inversion

Round-Trip Consistency

Applying the tensorial rule to transform a tensor from a source chart to a target chart, and then applying it again to transform back from the target chart to the source chart, recovers the original components exactly:

\sum_{j}^{n} {(J^{- 1})}_{i}^{j} J_{j}^{k} = δ_{i}^{k}

which is precisely the Jacobian product identity, and its role here is to guarantee that the tensorial rule is compatible with reversing the direction of any coordinate change, not merely with proceeding forward through a chain of charts.

Compatibility With the Identity Transformation

Trivial Case as a Consistency Check

When the source and target charts coincide, the transition map reduces to the identity map, whose Jacobian is the identity matrix at every point, and the tensorial rule applied with this trivial Jacobian correctly leaves every tensor component unchanged, confirming that the rule does not introduce any spurious change when no actual coordinate change has taken place.

Anchoring Point for the General Rule

This identity case serves as a base case against which the general compatibility of the rule under composition can be checked, since composing any transformation with its own inverse must reduce to this identity case by the round-trip consistency property described above.

Diagram of Compatibility Across a Chain of Charts

Three Charts, Two Routes, One Result

Why Compatibility Cannot Be Assumed Without Proof

The Role of the Chain Rule

Compatibility across composed transformations is not a separate postulate added on top of the tensorial rule; it follows automatically from the ordinary chain rule of multivariable calculus applied to the composition of the underlying transition maps, meaning that as long as the transition maps themselves are smooth enough to be differentiated the required number of times, the tensorial rule is automatically compatible with any composition built from them.

Failure Under Insufficient Smoothness

If a transition map fails to be sufficiently smooth at some point, differentiating it there may not be valid, and the chain rule argument underlying compatibility breaks down at that point, meaning the tensorial rule cannot be relied upon to give a consistent result for a chain of transformations passing through such a point, which is why smoothness of every transition map involved is a standing requirement for the tensorial rule's coordinate change compatibility to hold.

Practical Significance

Freedom to Choose a Convenient Route

Because coordinate change compatibility is guaranteed, a computation involving several coordinate systems can freely choose whichever sequence of intermediate charts is most convenient, confident that the final transformed tensor components will agree regardless of the specific path taken through the available charts, a freedom that is routinely exploited when a direct transition map between two charts of interest is inconvenient to write down explicitly but an intermediate chart simplifies the computation considerably.