6.14.2 Tensor One One Contravariant Slot

Tensor One One Contravariant Slot is the single upper argument position possessed by a type one-one tensor, accepting one one-form, that when left open while the tensor's covariant slot is filled with a vector, presents the tensor's output as a vector read directly from this position, 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 covariant slot, this contravariant slot completes the two-slot structure defining the type, and it is specifically the contravariant slot that plays the role of presenting the operator's output.

The Slot as the Operator's Output Position

One Position Accepting a One-Form, or Left Open to Read a Vector

The contravariant slot of a type one-one tensor is the unique upper-indexed position in the tensor's definition as a bilinear map, and it accepts exactly one one-form when the tensor is fully evaluated to a scalar. When instead the tensor's covariant slot has already been filled with a vector, leaving the contravariant slot open, the resulting partially evaluated object is itself a vector, with the contravariant slot serving as the position from which that output vector's components are read.

Distinguishing Output Presentation From Full Evaluation

Leaving the contravariant slot open, with the covariant slot already filled, is different from fully evaluating the tensor, which would additionally require a one-form to be supplied to this contravariant slot as well, collapsing the result to a scalar. The contravariant slot's role as an output-presenting position depends precisely on this partial evaluation being treated as meaningful in its own right, retaining an open slot rather than demanding immediate reduction to a number.

Components Associated With the Slot

One Upper Index Labels the Slot

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

Transformation Governed by Direct-Jacobian Behavior

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

The Slot in Relation to the Covariant Slot

Independence Between the Two Slots

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

The Two Slots Together Define the Operator's Full Action

While the contravariant slot alone presents the operator's output, the complete characterization of that output as a genuine vector relies jointly on the covariant slot having already received the operator's input: it is only because the covariant slot has been filled that the object read from the open contravariant slot carries the meaning of the operator's result on that particular input, rather than remaining an unevaluated tensor with two slots still open.

Behavior of the Contravariant Slot Under Operations

The Slot as a Source 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 contravariant slot that supplies the intermediate result passed into the second tensor's input, linking the two operators together into a single composed operator whose own contravariant slot presents the final output of the entire composition.

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, the contravariant slot, once the covariant slot has received any vector, presents that identical vector unchanged, illustrating in the simplest possible instance how the contravariant slot delivers the operator's output without alteration from whatever vector was supplied to the covariant slot.