3.5.4 Tensor Basis Covector Coordinate Extraction

Extracting covector coordinates from a tensor basis involves mapping dual space elements to coordinate systems through linear functionals.

Tensor Basis Covector Coordinate Extraction is the specific procedure for finding the coordinates of one fixed basis covector e^i, itself an element of V*, when re-expressed relative to a second, different dual basis {e'^k}, and the confirmation that two independent routes to this result, direct evaluation against the second basis and the general covariant transformation rule, necessarily agree.

Setting Up the Extraction Problem

A Basis Covector Is Still Just a Covector

Although e^i is singled out as a member of a particular dual basis, it remains, structurally, an ordinary element of V*, and so it has its own well-defined coordinates relative to any other basis of V*, exactly as any covector does. Given a second basis {e'_k} of V, with dual basis {e'^k}, the coordinates of e^i relative to {e'^k} are the scalars c_k in the expansion e^i = Σ_k c_k e'^k.

Method One: Direct Evaluation

Applying the General Extraction Formula

By the general coordinate-extraction formula for a covector, c_k = e^i(e'_k), since the k-th coordinate of any covector relative to {e'^k} is obtained by evaluating that covector on the k-th basis vector e'_k of the corresponding primal basis. Expanding e'_k = Σ_j a^j_k e_j in terms of the original basis and using e^i(e_j) = δ^i_j gives:

c_{k} = e^{i} ({e^{'}}_{k}) = e^{i} (\sum_{j}, a_{k}^{j}, e_{j}) = \sum_{j} a_{k}^{j} e^{i} (e_{j}) = a_{k}^{i}

so the coordinates of e^i relative to {e'^k} are exactly the entries a^i_k of the transition matrix from {e_j} to {e'_k}.

Method Two: The General Covariant Transformation Rule

Applying the Standard Covector Transformation

Treating e^i as a covector with coordinates ω_j = δ^i_j relative to the original dual basis {e^j}, the general covariant transformation rule, ω'_k = Σ_j a^j_k ω_j, gives:

c_{k} = \sum_{j} a_{k}^{j} δ_{j}^{i} = a_{k}^{i}

since the Kronecker delta collapses the sum to the single term with j = i, giving a^i_k, exactly matching the result of the direct evaluation method.

Agreement Between the Two Methods

Why the Two Routes Must Coincide

Both methods compute the same well-defined quantity, the coordinates of the fixed covector e^i relative to a second basis, so their agreement is not a coincidence but a necessary consequence of the coordinate transfer rules being derived, in the first place, from exactly the same evaluation principle used in method one; the general transformation rule for covector components is itself proved by an argument structurally identical to the direct evaluation computed here, applied to an arbitrary covector rather than the specific covector e^i.

The Result in Terms of the Original Transition Data

The clean outcome, that e^i's coordinates relative to {e'^k} are exactly the column of the transition matrix A associated with e_i mapped through, a^i_k, means that the coordinate array of a basis covector under a change of dual basis is read directly off the same transition matrix already used to relate the two primal bases, with no separate matrix computation required beyond A itself.

Special Case: Extraction Relative to the Covector's Own Dual Basis

Recovering the Trivial Kronecker Result

If {e'^k} is taken to be the same dual basis as {e^i} itself, A = I, the formula gives c_k = δ^i_k, recovering the trivial statement that e^i's coordinates relative to its own dual basis are 1 in position i and 0 elsewhere; this consistency check confirms the general extraction formula reduces correctly to the Kronecker relation in the degenerate case of no actual change of basis.