4.3.3 Tensor Trilinear Third Argument Slot

The trilinear third argument slot positions the third vector in a trilinear map, enabling operations across three vector spaces.

Tensor Trilinear Third Argument Slot is the designated input position of a trilinear map that, when the first and second arguments are both held fixed at any chosen values, receives a linear function of that single remaining variable. Given a trilinear map $T : V \times W \times X \to U$ the third argument slot is the coordinate reserved for elements of $X$ , and trilinearity guarantees that fixing any $v \in V$ and $w \in W$ produces a linear map $T (v, w, \cdot) : X \to U$ acting purely on the third slot.

Formal Statement

The Linearity Condition

For every fixed pair $v \in V$ , $w \in W$ , every $x_{1}, x_{2} \in X$ , and every scalar $γ$ , the third argument slot satisfies

T (v, w, x_{1} + γ x_{2}) = T (v, w, x_{1}) + γ T (v, w, x_{2})

Notation for the Frozen Partial Map

The partial map obtained by freezing the first and second slots is written

T_{v, w} : X \to U, T_{v, w} (x) = T (v, w, x)

The third slot completes the family of three partial linear maps that together characterize a trilinear map, alongside the analogous constructions for the first and second slots.

Coordinate Description

Index Position in the Structure Constants

Relative to bases with structure constants $c_{i j k}^{l}$ , the third argument slot corresponds to the index $k$ . Fixing $i$ and $j$ leaves an expression linear in $x^{k}$ :

T (v, w, x) = \sum_{k} x^{k} (\sum_{i, j} c_{i j k}^{l} v^{i} w^{j}) u_{l}

Currying and Embedding

Embedding into the Last Tensor Factor

Under the universal property of the triple tensor product, the third argument slot's linearity is what allows the assignment $x \mapsto v \otimes w \otimes x$ , for fixed $v$ and $w$ , to be linear, identifying the third slot with the rightmost tensor factor in $V \otimes W \otimes X$ .

Reduction to a Bilinear Map

Fixing only the third slot at a specific value $x_{0}$ yields a bilinear map $T (\cdot, \cdot, x_{0}) : V \times W \to U$ , showing that evaluating the third slot alone reduces trilinear structure to bilinear structure in the remaining two arguments, the natural counterpart to fixing the first slot to reduce to a bilinear map in the second and third.

Significance in Applications

The Trailing Slot in Composed Operations

In many concrete trilinear maps, such as the scalar triple product $v \cdot (w \times x)$ , the third argument enters through an operation, here the cross product, applied to the second and third slots jointly before combining with the first; even though the third slot's individual linearity still holds by trilinearity of the overall expression, recognizing which slot plays this trailing computational role is essential for correctly evaluating and differentiating such expressions.

Contraction on the Last Index

Contracting a rank-three tensor against a covector in its third index produces a rank-two tensor by evaluating the third argument slot at a fixed vector while leaving the first and second slots free. Because index order determines which contraction is performed, precise identification of the third argument slot, as distinct from the first and second, is necessary whenever the tensor's components are not fully symmetric.