4.2.2 Tensor Bilinear Second Argument Slot

The Tensor Bilinear Second Argument Slot is the second input in a bilinear map, combining vectors into higher-dimensional objects via bilinear operations.

Tensor Bilinear Second Argument Slot is the second of the two ordered argument positions in a rank-2 tensor's bilinear map presentation, distinguished by its own type, its own linearity behavior, and its own transformation rule under change of basis or under pullback and pushforward by a linear map, independently of the first slot.

Position and Typing

Identifying the Second Slot

For a bilinear map B(x, y), the second slot is the position occupied by y. Its type, either V or V*, is fixed by the tensor's valence: for a type (0, 2) tensor the second slot accepts a vector, for a type (2, 0) tensor it accepts a covector, and for a type (1, 1) tensor written as T(ω, v) the second slot conventionally accepts the vector argument while the first accepts the covector.

T : (\underset{first slot}{X}, \underset{second slot}{Y}) \mapsto T (x, y)

Independence from the First Slot's Type

The type of the second slot is not constrained by the type of the first: a bilinear map can pair a covector in the first slot with a vector in the second, or two covectors, or two vectors, and each combination corresponds to a genuinely different tensor valence. This independence is what allows mixed tensors, where the two slots have different types, to exist alongside purely covariant or purely contravariant rank-2 tensors.

Linearity in the Second Slot

The Second-Slot Linearity Condition

Holding the first argument x fixed, the map y ↦ T(x, y) is required to be linear:

T (x, a y_{1} + b y_{2}) = a T (x, y_{1}) + b T (x, y_{2})

This condition holds for every fixed x, and it is exactly what allows the second slot, on its own, to be treated as a linear functional once the first slot has been saturated: fixing x defines an element T(x, ·) of the dual space of whichever space the second slot draws from.

Currying on the Second Slot

Fixing the first argument produces a linear map depending only on the second slot,

T (x_{0}, \cdot) : Y \to ℝ

and this currying of the second slot is the mechanism by which a bilinear tensor of type (0, 2) defines a linear map V → V*, sending each x_0 to the functional T(x_0, ·); this map is precisely the lowering-index operation associated to the bilinear form when it is symmetric and nondegenerate.

Coordinate Behavior of the Second Slot

Index Assigned to the Second Slot

In component notation, the second slot corresponds to the second index of the tensor's component array. For a type (0, 2) tensor this is the second lower index, T_ij, with j ranging over the basis used to expand the second-slot argument:

T (u, v) = T_{ij} u^{i} v^{j}

Reordering the two slots, when they carry the same type, corresponds to transposing this index, and for a non-symmetric tensor T_ij ≠ T_ji in general, so the second slot's index position must be tracked consistently to recover the correct value.

Transformation Under Change of Basis

If the basis changes by an invertible matrix A, the second-slot index transforms by the same rule appropriate to its type: for a lower (covariant) second slot, components transform with a factor of A, matching how components of the second argument itself transform when re-expressed; for an upper (contravariant) second slot, the transformation uses the inverse matrix. This ensures the value T(u, v) is invariant under the simultaneous change of basis in both the tensor's components and the argument's coordinates.

Second Slot Under Pullback

Transformation Rule for a Covariant Second Slot

Given a linear map f: V → W and a type (0, 2) tensor T on W, the pullback f*T is defined by acting with f on both slots, including the second:

(f^{*} T) (u, v) = T (f (u), f (v))

so the second-slot argument v is mapped forward by f before being handed to T, exactly mirroring the treatment of the first-slot argument, which is the reason pullback of a symmetric or antisymmetric bilinear tensor preserves that symmetry: both slots are transformed by the identical map f.

Asymmetric Treatment in Mixed Tensors

For a type (1, 1) tensor T(ω, v) with the second slot accepting a vector, pullback along an invertible f transforms the second slot by f itself (pushforward-style), while the first slot, accepting a covector, transforms by (f^-1)*, illustrating that the second slot's transformation rule under a map is governed entirely by its own type, independent of how the first slot is transformed.