4.10.3 Tensor Vector Valued Component Output

Tensor Vector Valued Component Output refers to the structured representation of tensor components as vector-valued outputs in algebraic computations.

Tensor Vector Valued Component Output is the individual scalar coordinate, obtained relative to a chosen basis of the codomain space, that results from projecting a vector-valued tensor's full output onto a single basis direction of that codomain. It is the elementary numeric quantity that, collected across every basis direction of the codomain, reconstitutes the complete vector valued output, and it is the piece of data actually stored or computed when a vector-valued tensor is represented numerically.

Formal Definition

Extracting a Single Coordinate

Given a vector-valued multilinear map $T : V_1 \times \cdots \times V_k \to W$ and a chosen basis ${f_1, \ldots, f_m}$ of $W$, the vector valued component output at basis direction $a$, for a given input tuple, is the coefficient $T_a(v_1, \ldots, v_k)$ appearing in the expansion

T (v_{1}, \dots, v_{k}) = \sum_{a = 1}^{m} T_{a} (v_{1}, \dots, v_{k}) f_{a}

Each $T_a$, considered on its own, is a scalar-valued multilinear map on the same domain, obtained by pairing the full vector valued output against the $a$-th element of the dual basis of $W^{*}$: $T_a = f^{a} \circ T$.

Full Component Array

Combining the codomain index $a$ with the ordinary domain indices produces the complete component array of the vector-valued tensor,

T_{j_{1} \dots j_{s}}^{a; i^{1} \dots i^{r}} = T_{a} (e^{i^{1}}, \dots, e_{j_{s}})

and this array is precisely the full listing of every vector valued component output across every basis combination of the domain and every basis direction of the codomain simultaneously.

Properties of Component Outputs

Each Component Is a Full Scalar-Valued Tensor

Because $T_a$ is obtained by fixing the codomain projection while leaving the domain arguments free, each individual component output, considered as a map of the original arguments, is itself a complete scalar-valued multilinear map of the same type $(r,s)$ as the original vector-valued map, satisfying slotwise linearity in every one of its domain slots on its own.

Independence and Interdependence

The $m$ component maps $T_1, \ldots, T_m$ are, as functions, algebraically independent of one another in general: no relationship between them is forced merely by the vector-valued structure. In applications with additional geometric constraints, such as symmetry of the torsion tensor's antisymmetric part, relationships among the $T_a$ may be imposed externally, but this is never automatic from the bare definition of a vector valued tensor.

Transformation Under Codomain Basis Change

Choosing a different basis ${f_1', \ldots, f_m'}$ for $W$, related to the original by an invertible matrix $B$, mixes the component outputs linearly: the new component functions are linear combinations of the old ones, determined entirely by $B$, while the underlying full vector valued output, and each of its evaluations, remains unchanged as an abstract element of $W$.

Computational Role

Numeric Representation of Vector-Valued Tensors

In any explicit numeric computation, a vector-valued tensor's value is invariably represented as its full array of component outputs relative to some fixed basis; the abstract vector valued output is never manipulated directly in a symbolic or numeric computation, only through this finite collection of scalar component outputs.

Reassembly Into the Full Output

Given all $m$ component outputs for a specific input tuple, the full vector valued output is recovered exactly by the basis expansion formula, summing each component multiplied by its corresponding basis vector of $W$; no information is lost in passing between the full vector valued output and its complete set of component outputs relative to a fixed basis.

Summary of Key Points

A vector valued component output is the scalar coefficient obtained by projecting a vector-valued tensor's full output onto one basis direction of the codomain.
Each component output, viewed as a function of the domain arguments, is itself a complete scalar-valued multilinear map of the same type.
The component maps are algebraically independent unless additional structure imposes relationships among them.
Changing the codomain basis mixes the component outputs linearly without altering the underlying abstract vector valued output.
The full set of component outputs relative to a fixed basis is what is actually stored and manipulated in any explicit numeric or symbolic computation.