10.15.1 Tensor Coordinate Transformation Source Chart

Tensor Coordinate Transformation Source Chart is the originating coordinate system, together with its domain of definition, in which tensor components are already known before a coordinate transformation process begins, serving as the fixed starting reference from which every new component in the target chart is computed.

Defining the Source Chart

Chart as a Coordinate Map

A chart assigns an ordered set of real-number coordinates to each point of some region, and the source chart is the specific chart whose coordinates $$x^i$$ hold the tensor components at the start of the transformation, prior to any conversion into the coordinates $${\overline{x}}^j$$ of the target chart.

Domain of the Source Chart

The source chart is defined only over some open region of the underlying space, and the tensor components it carries are meaningful only at points within that domain, so the coordinate transformation process can only be initiated at points where the source chart is actually defined.

Role in the Transformation Formulas

Anchoring the Summed Index

Every index that is summed over in a transformation formula, rather than left free, refers to a direction in the source chart, so that in an expression such as:

the index $$i$$ ranges over the coordinate directions of the source chart, and the component $$W_i$$ is understood to be given data, already expressed with respect to the source chart's coordinate basis, before the transformation is applied.

Supplying the Basis at the Source

The source chart determines its own coordinate basis vectors $$e_i$$ at every point of its domain, and the tensor's original components are precisely the coefficients that express the tensor, as a fixed geometric object, in terms of this source basis; the entire purpose of the transformation is to re-express the same object using the target chart's basis instead.

Distinguishing Source From Target

Not an Inherent Property of a Chart

No chart is intrinsically a "source" or a "target"; a chart earns the label of source chart only relative to a specific coordinate transformation process, and the very same chart can serve as the target chart of a different transformation running in the opposite direction, where the roles of forward Jacobian and inverse Jacobian are correspondingly swapped.

Reversibility of the Roles

If the components known in the target chart are transformed back using the inverse Jacobian in place of the forward Jacobian, and vice versa, the original source chart becomes the new target chart, and the Jacobian product identity guarantees this reverse process recovers the original components exactly, confirming that the source and target labels are a matter of direction of travel rather than a fixed property.

Diagram of the Source Chart's Position

Starting Point of the Process

Requirements on the Source Chart

Smoothness and Non-Degeneracy

For the transformation process to be well defined, the source chart's coordinates must vary smoothly over its domain, and the transition map connecting the source chart to the target chart must be differentiable enough that all needed Jacobian and inverse Jacobian entries exist and are finite at the point under consideration.

Consistency of the Given Components

The tensor components given in the source chart must themselves be consistent with the tensor being a single well-defined geometric object, meaning that if the source chart is later replaced by an even earlier chart through a prior transformation, applying both transformations in sequence must agree with applying the single combined transformation directly, a requirement guaranteed by the chain rule for composed coordinate changes.

Multiple Source Charts for One Target

Overlapping Regions

When two different source charts each cover part of the same underlying region and both overlap with a single target chart's domain, tensor components transformed into the target chart from either source chart must agree on the overlap, since both describe the same underlying tensor field, and this agreement is what allows the various source charts to be stitched together into one consistent, globally defined tensor field on the target chart's domain.