11.13.1 Tensor Covariant Slot Vector Input

Tensor Covariant Slot Vector Input is the specific requirement that the argument accepted by a covariant slot of a tensor, when the tensor is viewed as a multilinear map, must be a contravariant vector rather than a covector, since only a vector supplies the correct type of object that a lower index position is structured to consume.

Definition and Basic Requirement

What Counts as a Valid Input

A covariant slot, corresponding to a lower index of the tensor, is defined to accept precisely a contravariant vector as its filling argument, meaning an object whose own components carry an upper index and transform according to the contravariant transformation law.

Why a Covector Cannot Be Substituted

Attempting to insert a covector, rather than a vector, into a covariant slot would require contracting two lower indices directly against one another without an intervening metric, which does not produce a coordinate-independent scalar and therefore fails to respect the pairing invariance that the slot structure is designed to guarantee.

Consistency With the Contraction Pattern

Matching Index Types in the Contraction

The vector input requirement for a covariant slot directly mirrors the general rule that a coordinate-independent contraction pairs one lower index with one upper index, since inserting a contravariant vector's upper-indexed components into the tensor's lower index produces exactly this required pairing.

Invariance of the Result Once the Correct Input Is Used

Because the vector input matches the covariant slot correctly, evaluating the tensor on this vector yields a scalar value, or a lower-rank tensor if other slots remain unfilled, that is guaranteed to be invariant under any change of coordinates, following directly from pairing invariance.

Behavior in Tensors With Several Covariant Slots

Each Slot Independently Requires a Vector

When a tensor carries multiple lower indices, each corresponding covariant slot independently imposes the vector input requirement, so a valid full evaluation of the tensor requires supplying one contravariant vector for every covariant slot present, with no slot accepting anything other than a vector.

No Substitution Across Slots

The vector input requirement applies uniformly to every covariant slot of the tensor, and there is no mechanism by which a covector could be substituted into one covariant slot while a vector fills another, since the requirement is fixed by the index type of each slot individually.

Role Within Tensor Algebras

Reinforcing the Definition of a Tensor as a Multilinear Map

The vector input requirement for covariant slots is a core part of what makes the multilinear map interpretation of tensors precise, specifying exactly which type of mathematical object belongs in each argument position rather than leaving the interpretation ambiguous.

Practical Guide for Constructing Valid Contractions

Recognizing that covariant slots require vector inputs serves as a practical check when constructing tensor expressions, helping to confirm that indices are being contracted correctly, with covariant slots always paired against contravariant vectors rather than against other covariant objects.