12.9.4 Tensor Evaluation Tensor Result Case

Tensor Evaluation Tensor Result Case is the situation in which a tensor is evaluated on fewer vectors and covectors than its full rank requires, producing not a single scalar but another tensor of reduced rank that still depends multilinearly on whatever arguments remain unfilled.

Setting Up Partial Evaluation

Filling Only Some of the Arguments

For a tensor $$A$$ of type $$(p,q)$$, which as a multilinear map accepts $$p$$ covectors and $$q$$ vectors, supplying only some of these arguments, such as a single vector $$v$$ in place of one of the $$q$$ vector slots while leaving the remaining slots unfilled, produces a new object:

where the dots denote slots that remain open, ready to accept further vectors or covectors.

The Result Is Itself a Tensor

This partially evaluated object is not a scalar, since it still awaits further multilinear inputs to fully saturate it. Instead, it behaves as a tensor of type $$(p,q-1)$$, having lost exactly one covariant slot corresponding to the argument that was filled.

Componentwise Description

Fixing One Index Through Contraction

In component form, supplying a specific vector $$v$$ for one of the tensor's covariant slots corresponds to contracting the tensor's components with the components of $$v$$ along that particular index, leaving the remaining indices free:

The resulting array $$B$$ has one fewer covariant index than $$A$$, and this reduction reflects precisely the loss of the filled slot.

Verifying Multilinearity Is Retained

Linearity in the Remaining Slots

The partially evaluated result remains multilinear in every one of its still-open slots, since fixing one argument of a multilinear map and leaving the others free produces a function that is still linear in each of those remaining arguments individually.

Linearity with Respect to the Filled Argument

The partially evaluated tensor also depends linearly on the vector or covector that was supplied to fill the slot, since substituting a linear combination of vectors into that slot produces the corresponding linear combination of the resulting partially evaluated tensors.

Distinguishing the Two Evaluation Cases

Full Evaluation Yields a Scalar

When every open slot of a tensor of type $$(p,q)$$ is filled with an appropriate vector or covector, the tensor evaluation operation produces a single scalar, with no remaining open slots.

Partial Evaluation Yields a Tensor of Lower Rank

When only some slots are filled, the result retains exactly as many open slots as remain unfilled, and this remaining structure is precisely what constitutes the tensor result case, distinguishing it from full evaluation which reduces entirely to a scalar.

Illustration