6.14.1 Tensor One One Covariant Slot

Tensor One One Covariant Slot is the single lower argument position possessed by a type one-one tensor, accepting one vector, that when filled leaves the tensor's upper slot open and thereby produces a new vector, this behavior being precisely what allows a type one-one tensor to act as a linear operator sending vectors to vectors. Alongside the tensor's lone contravariant slot, this covariant slot completes the two-slot structure defining the type, and it is specifically the covariant slot that plays the role of receiving the operator's input.

The Slot as the Operator's Input Position

One Position Accepting a Vector

The covariant slot of a type one-one tensor is the unique lower-indexed position in the tensor's definition as a bilinear map, and it accepts exactly one vector. Supplying a vector to this slot alone, while leaving the tensor's contravariant slot unfilled, does not immediately produce a scalar, since one slot remains open; instead it produces a new object with a single open upper slot, which is precisely a vector.

Distinguishing Input Reception From Full Evaluation

Filling only the covariant slot, leaving the contravariant slot open, is different from fully evaluating the tensor, which would additionally require a one-form to be supplied to the contravariant slot as well. The covariant slot's role as an input-receiving position depends precisely on this partial evaluation being treated as meaningful in its own right, producing a vector rather than demanding immediate reduction to a scalar.

Components Associated With the Slot

One Lower Index Labels the Slot

When expressed through components in a chosen basis, the covariant slot corresponds to exactly one lower index on the tensor's component array, distinct from the single upper index associated with the tensor's other slot. This lower index ranges over every basis direction of the underlying vector space, exactly as the lower index of a type zero-one tensor would.

Transformation Governed by Inverse-Jacobian Behavior

The covariant slot contributes exactly one factor of the inverse Jacobian matrix to the tensor's overall transformation law, this factor acting purely on the lower index and entirely independently of the direct-Jacobian factor separately contributed by the upper index.

The Slot in Relation to the Contravariant Slot

Independence Between the Two Slots

Because the covariant slot and the contravariant slot of a type one-one tensor are distinct positions accepting different kinds of argument, filling one has no direct effect on how the other must be filled; a vector supplied to the covariant slot and a one-form supplied to the contravariant slot are entirely independent choices, and the multilinearity of the tensor guarantees the final scalar depends linearly and separately on each.

The Two Slots Together Define the Operator's Full Action

While the covariant slot alone receives the operator's input, the complete characterization of the operator's output as a vector relies jointly on the presence of the open contravariant slot: it is only because the contravariant slot remains available after the covariant slot is filled that the partially evaluated object retains the character of a vector rather than collapsing directly to a number.

Behavior of the Covariant Slot Under Operations

The Slot as a Target for Contraction in Composition

When two type one-one tensors are composed by contracting the contravariant slot of one against the covariant slot of the other, it is specifically this covariant slot that receives the contracted index from the first tensor's output, linking the two operators together into a single composed operator whose own covariant slot is inherited unchanged from whichever tensor supplies the initial input.

The Slot's Role in the Kronecker Delta

In the particular case of the Kronecker delta, the type one-one tensor representing the identity operator, filling the covariant slot with any vector returns that identical vector from the open contravariant slot, illustrating in the simplest possible instance how the covariant slot channels an input vector through to the operator's output without alteration.