4.13.5 Tensor Additivity Component Form

Tensor Additivity Component Form is the expression of the tensor multilinear additivity property directly in terms of the numerical coordinates of an argument once a basis has been fixed, showing that additivity in a slot corresponds exactly to the ordinary coordinatewise addition of that argument's components. It translates the abstract, basis-independent additivity identity into a concrete statement about how sums of coordinate arrays behave when substituted into the component evaluation formula for a tensor.

Setting Up the Component Form

Coordinates of a Sum

Let V be a finite-dimensional vector space with basis e_1, ..., e_n. If two vectors u and w have coordinates u^j and w^j respectively relative to this basis, their sum u + w has coordinates equal to the ordinary sum of the two coordinate arrays:

This is simply the definition of vector addition expressed in coordinates, and it is the starting point for translating additivity into a statement purely about numbers.

Substituting the Summed Coordinates

For a type (p, q) tensor T with components T^{i_1 ... i_p}_{j_1 ... j_q}, substituting the coordinates of u + w into the l-th covariant slot of the component evaluation formula gives, for that slot's index j_l, the coordinate u^{j_l} + w^{j_l} in place of a single unsummed coordinate.

The Additivity Identity in Coordinates

Distributing the Sum Through the Component Formula

Because ordinary multiplication distributes over addition, the term T^{i_1 ... i_p}_{j_1 ... j_l ... j_q} (u^{j_l} + w^{j_l}), appearing inside the summation over j_l, splits into two separate terms:

Carrying this splitting through the entire summation over every index reproduces exactly the additivity identity T(..., u + w, ...) = T(..., u, ...) + T(..., w, ...), now derived purely from the distributive law of ordinary arithmetic applied to the component sum.

Additivity as a Property of Linear-in-Coordinate Formulas

More generally, the component evaluation formula exhibits additivity in slot l precisely because the formula is linear, to the first power, in the coordinate v_l^{j_l} associated with that slot: any formula in which the coordinates of one argument appear multiplied together, rather than singly, would fail to satisfy this distributive splitting and would therefore fail additivity in that slot.

Verifying Additivity by Inspecting Components

The Linear-in-Coordinates Criterion

Given an explicit component formula for a candidate multilinear map, additivity in a given slot can be verified directly by checking that the formula involves the coordinate associated with that slot exactly once, appearing only as a factor multiplied by other terms, and never combined with itself through squaring, multiplication by another coordinate from the same argument, or any other nonlinear operation.

Example of a Formula Satisfying Additivity

The bilinear form component formula ∑_{i,j} T_{ij} v^i w^j satisfies additivity in its second argument because w^j appears only to the first power and only multiplied by the fixed quantities T_{ij} v^i; replacing w with w_1 + w_2 and expanding via the distributive law directly reproduces the sum of the two separate evaluations.

Example of a Formula Failing Additivity

A formula such as ∑_i T_i (v^i)^2, in which the coordinate of the single argument appears squared, fails additivity in that argument, since (u^i + w^i)^2 does not equal (u^i)^2 + (w^i)^2 in general; such a formula does not describe a multilinear map and therefore cannot correspond to any tensor.

Component Additivity and Basis Independence

The Same Additivity Holds in Every Basis

Although the component form of additivity is expressed relative to a specific basis, the underlying additivity property of T itself does not depend on that basis; expressing the same tensor in a different basis changes the numerical values of the components and coordinates involved, but the distributive splitting that establishes additivity continues to hold identically, since it follows from the same underlying linear-in-coordinate structure in any basis.

Consistency with Basis-Free Additivity

The component form of additivity is not a separate or weaker version of the basis-free additivity identity; it is the same identity, made concrete by substituting specific numerical coordinates, and the two formulations agree exactly once the coordinate expansions of the abstract vectors are substituted into the abstract identity.

Diagrammatic Summary

The diagram shows how the summed coordinate u^j + w^j, substituted into the component evaluation formula, distributes through the sum to yield exactly the two separate component evaluations corresponding to u and to w.