10.23.4 Tensor Change of Basis Coordinate Transformation Boundary

Tensor Change of Basis Coordinate Transformation Boundary is the specific limitation, arising from the properties of the map connecting two coordinate systems, that restricts the region over which that transformation can be used to convert tensor components between the two systems, distinct from limitations belonging to either coordinate system considered on its own.

The Transformation Map as the Source of the Boundary

Definition Through a Coordinate Map

A change of basis is generated by a coordinate transformation, a function expressing the coordinates of one system as functions of the coordinates of another. The transformation boundary is the locus, within the common domain of the two coordinate systems, where this function fails to have the properties required for tensor components to be converted correctly.

Required Properties Away From the Boundary

Within the region where the transformation is well behaved, the map must be smooth enough to differentiate as many times as the calculation requires, and its Jacobian matrix must be invertible, so that both the direct and inverse Jacobian factors needed for tensor transformation exist and are finite.

Types of Transformation Boundaries

Non-Invertibility Boundary

The most fundamental boundary occurs where the Jacobian determinant of the transformation vanishes. At such points the map locally collapses dimensions, several source points may map to the same target point, or the inverse map may fail to be single valued, so tensor components cannot be converted consistently.

Smoothness Boundary

Even where the transformation is invertible, it may fail to be differentiable as many times as required, for instance being continuous but not smooth at a particular locus. Wherever the needed order of derivative fails to exist, the corresponding Jacobian factor, or a derivative of it required by a more advanced formula, is undefined at that locus, marking a smoothness boundary distinct from an invertibility boundary.

Domain Boundary Set by the Map's Own Formula

Some transformation formulas are simply not defined outside a certain range of input values, for example, a formula involving a square root or a logarithm that requires its argument to stay positive. The set of coordinates violating this requirement lies outside the domain of the transformation itself, forming a boundary imposed directly by the algebraic form of the map.

Consequences for Tensor Components

Discontinuity or Blow-Up of Transformed Components

As a point approaches a transformation boundary from within the valid domain, the transformed tensor components may grow without bound, oscillate, or fail to approach a limit, even if the original, untransformed components remain perfectly finite and smooth at the corresponding point.

Restriction of Validity for Derived Formulas

Any identity or formula derived using the direct or inverse Jacobian factors of a particular transformation, such as a relation between the volume elements of two coordinate systems, inherits the transformation boundary as part of its own domain of validity, and citing such a formula outside that domain produces a result with no guaranteed meaning.

Relationship to Composed Transformations

Boundaries Compound Under Composition

When two coordinate transformations are composed to relate a chain of three coordinate systems, the resulting boundary of validity is the union of the boundaries of each individual transformation together with any new degeneracy introduced by the composition itself, so a composed transformation can have a smaller domain of validity than either transformation taken alone.

Verifying the Composed Domain Explicitly

Because boundaries compound in this way, a chain of changes of basis must have its overall domain of validity checked explicitly, rather than assumed from the validity of each individual step, since a point can lie safely inside the domain of every individual transformation while still lying on the boundary of the composition as a whole.

Practical Identification of the Boundary

Direct Computation of the Jacobian Determinant

The most direct way to locate a transformation boundary is to compute the Jacobian determinant symbolically and identify the locus where it vanishes, together with any locus excluded by the algebraic domain of the transformation formula itself.

Cross-Checking With a Third Coordinate System

Where it is unclear whether an apparent boundary reflects a genuine limitation of the transformation or merely an artifact of how the formula happens to be written, expressing both coordinate systems in terms of a third, independently well-behaved coordinate system and comparing results is a reliable way to confirm the true extent of the transformation boundary.