10.15.3 Tensor Coordinate Transformation Map

The Tensor Coordinate Transformation Map explains how tensors adapt under coordinate changes while maintaining their geometric meaning across different systems.

Tensor Coordinate Transformation Map is the underlying smooth function relating the coordinates of the source chart to the coordinates of the target chart, prior to any differentiation, whose existence and invertibility are the precondition that makes it possible to build a Jacobian relation and carry out the transformation of tensor components at all.

Definition of the Map

Coordinate Functions

The transformation map is a collection of real-valued functions, one for each new coordinate, expressing every target coordinate as a function of all the source coordinates simultaneously:

{\bar{x}}^{j} = {\bar{x}}^{j} (x^{1}, \dots, x^{n})

Taken together over all values of $j$ , these functions constitute a single map sending a point described by source coordinates to the same point described by target coordinates.

Requirement of Smoothness

For the coordinate transformation process to define a Jacobian relation at all, the transformation map must be differentiable, and for the higher-order transformation laws used with tensors of rank two or more, it is generally required to be smooth to whatever order the application demands, since each additional derivative taken of the map may be needed when transforming connection coefficients or curvature-related objects built from second derivatives.

Invertibility of the Map

Local Inverse Function Theorem

The transformation map admits a local inverse, allowing the source coordinates to be recovered from the target coordinates, precisely at those points where the Jacobian matrix built from the map's partial derivatives has a non-zero determinant, a guarantee supplied by the inverse function theorem:

det (\frac{\partial {\bar{x}}^{j}}{\partial x^{i}}) \neq 0

Local Versus Global Invertibility

The inverse guaranteed by this theorem is only local: the transformation map may fail to be one-to-one across its entire domain even while remaining locally invertible near every individual point, so the coordinate transformation process is typically applied within a neighborhood small enough that the map behaves as a genuine bijection there.

The Map as the Source of the Jacobian

Differentiating the Map

The Jacobian matrix used throughout the coordinate transformation process is obtained by differentiating the transformation map's component functions, so the map itself is logically prior to the Jacobian: the Jacobian is a derived quantity, while the map is the primitive object relating the two coordinate systems as sets of points, not merely as linearized directions.

Second-Order Information Beyond the Jacobian

Some tensor-related quantities require not just the first derivatives collected in the Jacobian but the second derivatives of the transformation map as well, and these second derivatives are only available because the map itself was assumed smooth to a high enough order at the outset, illustrating why the map carries more information than the Jacobian relation alone.

Diagram of the Map

Points Carried Between Charts

Special Kinds of Transformation Maps

Linear Maps

When the transformation map is linear, it can be written directly as a constant matrix acting on the source coordinates, with no additional derivative computation needed since the Jacobian of a linear map equals the matrix of the map itself, independent of position.

Curvilinear Maps

When the transformation map is a general nonlinear function, such as one relating Cartesian coordinates to polar, cylindrical, or spherical coordinates, the Jacobian varies from point to point, and the coordinate transformation process must recompute the Jacobian relation freshly at every point where a tensor component is being transformed.

Composition of Maps

Chained Transformation Maps

If a transformation map is applied to go from a source chart to an intermediate chart, followed by a second transformation map from the intermediate chart to a target chart, the composition of the two maps is itself a valid transformation map directly relating the source chart to the target chart, and its Jacobian is given by the chain rule as the product of the two intermediate Jacobians.

Identity Map as a Trivial Case

When the source and target charts coincide, the transformation map reduces to the identity map, whose Jacobian is the identity matrix at every point, correctly reflecting that no actual change of tensor components should occur under a transformation that does not change the coordinate system at all.