9.6.4 Tensor Noncoordinate Basis Local Representation

Tensor Noncoordinate Basis Local Representation is the expression of each member of a noncoordinate frame as a linear combination of the vectors of an ordinary coordinate basis defined on the same neighborhood, with coefficients that vary smoothly from point to point, providing a concrete, calculable stand-in for the noncoordinate frame in terms of familiar coordinate quantities; it is what allows a noncoordinate basis, despite having no coordinate functions of its own, to be written down, differentiated, and computed with by referring back to some coordinate system already available on the same region.

Constructing the Local Representation

Expressing Frame Members as Coordinate Combinations

Given a coordinate basis {∂/∂x^i} on a neighborhood, a noncoordinate frame {e_a} defined on the same neighborhood is represented locally by writing each frame member as a combination of the coordinate basis vectors, with coefficients e_a^i that are themselves functions of position.

The Coefficient Matrix Must Be Invertible

For this local representation to define a genuine frame, spanning the tangent space at every point of the neighborhood, the matrix of coefficients e_a^i must be invertible at each point, guaranteeing that the frame members remain linearly independent throughout the region.

Recovering Structure Functions From the Local Representation

Brackets Computed From the Coefficient Functions

Because the local representation reduces each frame member to an explicit combination of coordinate basis vectors, the bracket of two frame members, and hence the structure functions of the noncoordinate basis, can be computed directly from the derivatives of the coefficient functions e_a^i, using the known vanishing bracket of the underlying coordinate vector fields.

The Representation Confirms the Frame Is Genuinely Noncoordinate

The vanishing or nonvanishing of the structure functions computed this way, from an explicit coefficient matrix, gives a concrete, checkable confirmation of whether the frame lies on the coordinate or noncoordinate side of the coordinate boundary, settling the question by direct computation rather than by abstract argument alone.

Passing Between Local Representations

Different Coordinate Backdrops Give Different Coefficient Matrices

The same noncoordinate frame can be given a local representation with respect to more than one underlying coordinate system, producing different coefficient matrices e_a^i in each case, related to one another by the ordinary Jacobian transformation between those coordinate systems.

The Frame Itself Is Independent of Which Backdrop Is Chosen

Although the coefficient matrix of the local representation depends on the coordinate backdrop chosen, the noncoordinate frame it represents is the same underlying object in every case, so any calculation carried out using one local representation must agree with the same calculation carried out using another, once translated through the appropriate Jacobian.

Diagram of a Local Representation

Consequences of the Local Representation

It Makes Noncoordinate Calculations Tractable

By reducing a noncoordinate frame to explicit coefficient functions over a known coordinate basis, the local representation allows differentiation, bracket computation, and component transformation involving the noncoordinate frame to be carried out using ordinary partial-derivative calculus, rather than requiring separate techniques for frames lacking coordinate origin.

It Separates the Frame's Identity From Any Particular Coordinate Choice

Because the same frame admits local representations with respect to different coordinate backdrops, results derived using one representation must be checked for consistency under change of backdrop; results that depend only on the frame itself, and not on the particular coordinate system used to represent it, are the ones properly attributable to the noncoordinate basis.