12.10.1 Tensor Substitution Target Slot

Tensor Substitution Target Slot is the specific argument position within a tensor's multilinear structure that has been designated to receive a particular vector or covector during slot substitution, identified precisely by its numbered position among the tensor's covariant or contravariant slots.

Identifying the Target Slot

Position Among the Tensor's Arguments

A tensor of type $$(p,q)$$ has a definite ordered list of argument positions: $$p$$ positions reserved for covectors and $$q$$ positions reserved for vectors. The target slot for a given substitution is one specific position among these, identified both by whether it accepts a vector or a covector and by its numbered place within that category.

Notation for the Target Slot

When a vector $$v$$ is substituted into the $$r$$-th vector slot of a tensor $$A$$, the target slot is denoted explicitly to avoid ambiguity:

with every other slot remaining an unfilled placeholder, distinct from the identified target.

Why Target Slot Identification Matters

Different Target Slots Generally Yield Different Results

Because a general tensor's components are not required to be symmetric across its indices, substituting the same vector into a different target slot typically produces a different resulting tensor, even though the same vector and the same original tensor are involved. Precisely identifying the target slot is therefore essential to determining which contraction is actually being performed.

Necessity for Unambiguous Composition

In a longer computation involving several slot substitutions performed one after another, correctly tracking which target slot each substitution addresses ensures that the sequence of operations corresponds to the intended contraction pattern, rather than an unintended rearrangement of which indices are contracted against which arguments.

Relationship to the Tensor's Index Structure

Correspondence Between Slot and Index

The target slot directly corresponds to one specific index of the tensor in its componentwise representation: a target vector slot corresponds to one covariant index, while a target covector slot corresponds to one contravariant index. Substitution into the target slot amounts to contracting precisely that corresponding index against the components of the supplied argument.

Renumbering After Substitution

Once a target slot has been filled through substitution, the resulting tensor of reduced rank has its remaining slots renumbered relative to the original tensor, since one position, the one identified as the target, no longer exists among the open slots of the new object.

Illustrative Example

Selecting the First Versus the Second Slot

For a rank-2 tensor $$A$$ with two vector slots, substituting $$v$$ as the target for the first slot produces a covector depending on whatever fills the second slot, while substituting $$v$$ as the target for the second slot instead produces a covector depending on whatever fills the first slot, and these two resulting covectors are generally different from one another.

Illustration