3.6.5 Tensor Covector Evaluation Coordinate Independence

Tensor covector evaluation remains consistent across coordinate systems, ensuring mathematical properties are preserved regardless of representation.

Tensor Covector Evaluation Coordinate Independence is the property that the scalar value produced by pairing a covector with a vector is the same real number regardless of which basis is used to represent the two objects in components, even though the individual component arrays f_i and v^i themselves change completely under a change of basis. This invariance is what elevates the evaluation operation from a mere formula in coordinates to a well-defined geometric pairing between the abstract spaces V* and V.

The Problem Coordinate Independence Solves

Components Are Basis-Dependent

Given a vector v in V, its component array (v^1, ..., v^n) depends entirely on the chosen basis; the same geometric vector has different numerical components in different bases. The same is true for a covector f, whose components (f_1, ..., f_n) also change with the basis of V* induced by the dual basis construction. Since both sets of components change, it is not obvious in advance that the sum f_i v^i computed in one basis will agree with the sum computed in another.

Why Invariance Is Required

For the evaluation operation f(v) to be a meaningful, basis-free definition, matching its abstract definition as f applied to v as a function, its component formula must produce the identical number no matter which basis was used to compute the components. Coordinate independence is the theorem guaranteeing this is always true.

Derivation of Coordinate Independence

Transformation of Vector Components

Suppose a new basis e'_1, ..., e'_n is related to the old basis e_1, ..., e_n by an invertible change-of-basis matrix A, with e'_j = A^i_j e_i. The components of a fixed vector v transform with the inverse matrix,

{\tilde{v}}^{i} = {(A^{- 1})}_{j}^{i} v^{j}

Transformation of Covector Components

The components of a fixed covector f transform with the matrix A itself,

{\tilde{f}}_{i} = A_{i}^{j} f_{j}

This opposite behavior, A^{-1} for vector components versus A for covector components, is exactly what the terms contravariant and covariant describe.

Cancellation in the Contracted Sum

Recomputing the evaluation in the new basis,

{\tilde{f}}_{i} {\tilde{v}}^{i} = A_{i}^{k} f_{k} {(A^{- 1})}_{j}^{i} v^{j}

The product (A^{-1})^i_j A^k_i sums to δ^k_j, since A^{-1} and A are inverse matrices, leaving

{\tilde{f}}_{i} {\tilde{v}}^{i} = δ_{j}^{k} f_{k} v^{j} = f_{j} v^{j}

which is identical to the value computed in the original basis, confirming that the scalar output is unchanged.

General Principle: Contravariant-Covariant Pairs Always Cancel

The Rule for Any Contraction

This cancellation is not special to the single covector-vector case; it is the general reason that any tensor contraction between an upper index and a lower index produces a basis-independent result. Whenever a change-of-basis matrix A acts on an upper index and its inverse A^{-1} acts on a paired lower index within the same contracted term, the two transformations compose to the identity, leaving the contracted quantity invariant.

Why Same-Variance Pairs Do Not Cancel

If two indices of the same variance were summed together, such as two upper indices or two lower indices, the transformation matrices would not be inverses of one another and no cancellation would occur; the resulting sum would depend on the basis and would not represent a legitimate coordinate-free tensor operation. This is why raising or lowering an index with a metric tensor is required before such same-variance quantities can be meaningfully combined.

Geometric Interpretation

Evaluation as an Intrinsic Pairing

Coordinate independence confirms that f(v) should be understood as an intrinsic pairing between the abstract vector space V and its dual V*, existing prior to and independently of any coordinate system. The choice of basis is merely a computational convenience for evaluating the pairing numerically; it never alters the geometric or algebraic content of the pairing itself.

Diagrammatic Summary

The diagram shows two different bases producing two different sets of components for the same covector and vector, both routes converging on the identical scalar output, illustrating coordinate independence directly.