3.4.3 Tensor Dual Basis Evaluation Rule

The Tensor Dual Basis Evaluation Rule explains how dual basis vectors interact with tensors to yield scalar values through precise algebraic operations.

Tensor Dual Basis Evaluation Rule is the general formula for computing the value ω(v) of an arbitrary covector on an arbitrary vector once both are expressed in coordinates relative to a basis and its dual basis, extending the elementary relation e^i(e_j) = δ^i_j, which applies only to basis vectors, to a fully general rule applicable to any vector and any covector whatsoever.

From Basis Vectors to General Vectors

The Elementary Case

The defining relation e^i(e_j) = δ^i_j specifies the dual basis vectors' action only on the basis vectors e_j themselves. Extending e^i linearly to a general vector v = Σ_j v^j e_j gives:

e^{i} (v) = e^{i} (\sum_{j}, v^{j}, e_{j}) = \sum_{j} v^{j} e^{i} (e_{j}) = \sum_{j} v^{j} δ_{j}^{i} = v^{i}

so e^i(v) = v^i, exactly the i-th coordinate of v, for every vector v, not merely for basis vectors.

The General Evaluation Formula

Pairing Two Coordinate Expansions

For a general covector ω = Σ_i ω_i e^i and a general vector v = Σ_j v^j e_j, the evaluation rule gives:

ω (v) = (\sum_{i}, ω_{i}, e^{i}) (\sum_{j}, v^{j}, e_{j}) = \sum_{i, j} ω_{i} v^{j} δ_{j}^{i} = \sum_{i} ω_{i} v^{i}

The double sum over i and j collapses to a single sum over one index once the Kronecker delta forces j = i; this collapse from a double sum to a single sum is the general evaluation rule, and it is the direct computational payoff of having a dual basis available at all.

Why This Formula Requires No Further Justification

Because the dual basis vectors were constructed precisely to satisfy e^i(e_j) = δ^i_j, and because both ω and v are, by definition of a basis, uniquely expressible as the coordinate sums used above, the general evaluation formula ω(v) = Σ_i ω_i v^i is not an additional assumption but a direct algebraic consequence of the dual basis definition together with bilinearity of the natural pairing.

Evaluation for Multilinear Tensors

Extension to Several Arguments

The same collapsing mechanism extends to higher tensors: for a (p, q)-tensor T with coordinates T^{i_1...i_p}_{j_1...j_q} relative to a basis and its dual, and arguments η^1, ..., η^p and u_1, ..., u_q expressed in the corresponding coordinates, evaluating T on these arguments produces a sum over all index combinations, weighted by products of the arguments' coordinates, with each factor of T's coordinate array contributing exactly as the basic evaluation rule prescribes for its own slot.

Practical Computation

In practice, evaluating a tensor on specific numerical arguments never requires re-deriving the dual basis relation from scratch; it is carried out by directly substituting the coordinates of the arguments into the general evaluation formula, Σ (index combination) T(...) · (argument coordinates), which is precisely the multi-index generalization of ω(v) = Σ_i ω_i v^i.

Consistency With Contraction

Evaluation as a Special Case of Contraction

The evaluation ω(v) = Σ_i ω_i v^i is the smallest possible instance of tensor contraction, pairing a single upper index, v^i, against a single lower index, ω_i; every more general contraction formula used elsewhere, for higher tensors or for partial evaluation of only some of a tensor's slots, applies this same evaluation rule slot by slot, confirming that the dual basis evaluation rule is the atomic operation from which all tensor contraction is built.