10.15.2 Tensor Coordinate Transformation Target Chart

Tensor Coordinate Transformation Target Chart is the destination coordinate system, together with its domain of definition and its compatibility conditions, into which tensor components are being transformed at the end of a coordinate transformation process, serving as the fixed reference against which every new component, basis vector, and transformation coefficient is expressed.

Defining the Target Chart

Chart as a Coordinate Map

A chart is a smooth, invertible map assigning an ordered set of real-number coordinates to each point of a region, and the target chart is the specific chart whose coordinates $${\overline{x}}^j$$ are being produced by the transformation, as opposed to the source chart whose coordinates $$x^i$$ are being left behind.

Domain of the Target Chart

The target chart is defined only over some open region of the underlying space, and the coordinate transformation process is valid only at points lying in the overlap between the source chart's domain and the target chart's domain, since outside that overlap either the original coordinates or the new coordinates, or both, are simply undefined.

Role in the Transformation Formulas

Anchoring the Free Index

Every free index left over after a transformation sum is completed refers to a direction in the target chart, so that a transformed component such as:

is understood as the component of the tensor with respect to the target chart's coordinate direction labeled by $$j$$, and this labeling only makes sense once the target chart has been fixed in advance.

Supplying the Basis at the Target

The target chart determines its own coordinate basis vectors $${\overline{e}}_j$$ at every point of its domain, and the transformed tensor components are precisely the coefficients needed to express the same tensor in terms of this target basis, so that the tensor, as a geometric object, remains unchanged while only its coordinate representation shifts to match the target chart.

Chart Overlap and Transition Maps

Transition Map Between Charts

When both the source chart and the target chart cover a common region, the relationship between them is described by a transition map, and the Jacobian and inverse Jacobian matrices used in the transformation are precisely the derivative data of this transition map, evaluated at the point being considered.

Consistency on Overlaps

If a third chart also covers part of the same region, the transition map from the source chart to the third chart must agree, on the overlap, with the composition of the transition map from the source chart to the target chart followed by the transition map from the target chart to the third chart, a compatibility condition guaranteed by the chain rule whenever all the transition maps are smooth.

Diagram of Source and Target Charts

Overlapping Domains

Choosing a Target Chart

Motivated Choices

The target chart is often chosen deliberately for a specific advantage, such as diagonalizing a metric tensor at a point, aligning coordinate lines with the symmetry of a physical system, or simplifying a differential equation, and the coordinate transformation process is the mechanism by which tensor components already known in the source chart are carried over to take advantage of that chosen target chart.

Multiple Valid Targets

A single source chart may be transformed into many different target charts for different purposes, and the transformation process treats each target chart independently, requiring its own Jacobian and inverse Jacobian matrices computed from the specific transition map connecting the source chart to that particular target chart.

Boundary Behavior at Chart Edges

Approaching the Domain Boundary

Near the boundary of the target chart's domain, coordinate lines may become degenerate or the Jacobian determinant may approach zero, signaling that the target chart is no longer suitable for representing the tensor faithfully in that region, and a further transformation to a different chart is required to continue past the boundary.

Charts That Cannot Cover Everything

Some underlying spaces cannot be covered by a single chart at all, meaning any target chart chosen will necessarily leave out some points, so the full description of a tensor field over the whole space may require an atlas of several overlapping target charts rather than one target chart alone.