11.5.3 Tensor Covariant Object Slot Input Role

Tensor Covariant Object Slot Input Role is the specific function performed by each covariant index slot of a tensor, namely accepting a contravariant vector as its argument, describing at the level of an individual slot rather than the whole object how a covariant tensor consumes vector inputs to build toward a final scalar output.

The Slot as an Input Receptacle

One Contravariant Vector per Covariant Slot

Each covariant slot of a tensor is designed to receive exactly one contravariant vector as input, with the slot's numerical action on that vector determined by the covariant components associated specifically with that slot.

Order Sensitivity of Slot Filling

Because slots are typically ordered, filling the first covariant slot with a given vector generally produces a different result than filling the second slot with the same vector, unless the tensor possesses a symmetry that makes the slots interchangeable, so the input role of a slot is tied specifically to its position among the tensor's covariant slots.

Sequential Filling of Multiple Slots

Reducing Rank One Slot at a Time

Supplying a vector to one covariant slot of a higher-rank covariant tensor consumes that slot and produces a new object of one lower covariant rank, which itself retains an input role toward its remaining slots, allowing the full contraction to a scalar to be understood as a sequence of single-slot input operations.

Independence of the Result From Filling Order When Symmetric

For a covariant tensor that is symmetric across two of its slots, the slot input role produces the same final number regardless of the order in which the two vectors are supplied, while for a general, non-symmetric tensor the order in which slots are filled must match the specific vectors intended for each slot.

The Slot Input Role Compared to the Contravariant Slot Output Role

Receiving Versus Supplying

A covariant slot's role is to receive a contravariant vector as input, in direct contrast to a contravariant slot on a mixed tensor, whose role is to receive a covariant object, such as a one-form, as its input; recognizing which type of object a given slot is designed to accept is essential for correctly applying a tensor of mixed type.

Consistency With the Overall Variance Type

The input role of every covariant slot on a given tensor is uniform in the sense that all such slots accept the same category of input, a contravariant vector, even though the specific numerical action of each slot differs according to its own set of covariant components.

Practical Use in Tensor Manipulation

Guiding Correct Contraction Order in Calculations

Thinking of covariant slots individually in terms of their input role clarifies, in a calculation involving several vectors and a higher-rank covariant tensor, exactly which vector must be supplied to which slot, reducing the risk of assigning a vector to the wrong slot when the tensor lacks a symmetry that would make the assignment order immaterial.

Basis for Defining Partial Contractions

The slot input role underlies the common practice of partially contracting a tensor, supplying vectors to only some of its covariant slots while leaving the remaining slots open, producing a new tensor of reduced rank that itself retains a well-defined input role for any slots not yet filled.