11.14.2 Tensor Contravariant Slot Upper Index Association

Tensor Contravariant Slot Upper Index Association links upper indices to coordinate transformations, defining how contravariant components are indexed in tensor algebra.

Tensor Contravariant Slot Upper Index Association is the direct correspondence, fixed by notational convention, between each upper index appearing on a tensor's symbol and one specific contravariant slot of that tensor viewed as a multilinear map, ensuring that the written position of an index unambiguously identifies which argument position expects a covector.

Definition and Notational Basis

The Correspondence Stated Precisely

Every distinct upper index written on a tensor's symbol is associated with exactly one contravariant slot, and the order in which upper indices are written from left to right is taken, by convention, to match the order in which the corresponding covector arguments are supplied to the multilinear map.

T^{i j} (ω, τ) = T^{i j} ω_{i} τ_{j}

Distinguishing Index Labels From Slot Identity

Although two different letters may be used to label two different upper indices, the association is with the slot's position, not the specific letter chosen, so relabeling an upper index with a different letter does not change which contravariant slot it identifies, provided its position among the indices is preserved.

Consequences of the Association

Enabling Unambiguous Partial Evaluation

Because each upper index is tied to a specific slot, a tensor with several upper indices can have just one of those slots filled with a covector while the others remain open, and the upper index association makes clear exactly which slot has been filled and which slots still await an argument.

T^{i j} ω_{i} = S^{j}

Consistency Requirement in Symmetric and Antisymmetric Tensors

For tensors with special symmetry properties among their upper indices, the upper index association must still be respected when applying the symmetry, since exchanging the roles of two contravariant slots is only meaningful once each slot's association with a specific index label has been correctly identified beforehand.

Interaction With Coordinate Transformation

Association Preserved Under a Change of Coordinates

When a tensor's components are transformed to a new coordinate system, the upper index association with its contravariant slots is preserved exactly, since the contravariant transformation law updates each index's numerical value while leaving unchanged which slot that index refers to.

Association Independent of the Specific Covector Supplied

The upper index association identifies a slot as a fixed position in the tensor's structure, entirely independent of which particular covector is eventually supplied to fill that slot, so the association remains stable even as different covectors are substituted into the same contravariant slot across different calculations.

Role Within Tensor Algebras

Supporting Precise Communication of Tensor Structure

The upper index association provides the precise bookkeeping needed to communicate unambiguously which contravariant slot of a tensor is being referenced in any given expression, which becomes essential once tensors possess more than one contravariant slot and careful tracking is required.

Foundation for Defining Operations on Specific Slots

Operations such as contraction, symmetrization, or index lowering that target one particular index of a tensor rely entirely on the upper index association to specify correctly which contravariant slot is being acted upon, distinguishing it from any other contravariant or covariant slot the tensor may possess.