9.18.2 Tensor Coordinate Free Map Description

Explore how tensor coordinate-free maps describe multilinear relationships without relying on specific coordinate systems.

Tensor Coordinate Free Map Description is the characterization of a tensor purely as a multilinear map acting on vectors and covectors, described entirely in terms of its input-output behavior and without any reference to coordinates, indices, or a chosen basis. It is the way of presenting what a tensor is by specifying what it does, rather than by specifying an array of numbers relative to some coordinate system.

The Map-Based Characterization

A Function of Several Arguments

A tensor of type (p, q) is described as a function that takes p covectors and q vectors as arguments and produces a single scalar, with the defining property that this function is linear separately in each of its arguments.

T : \underset{p}{\times} V^{*} \times \underset{q}{\times} V \to ℝ

No Coordinates Required to State the Definition

This description is complete without ever selecting a basis: the domain and codomain of the map are specified abstractly, and the multilinearity condition is stated purely in terms of vector space operations, addition and scalar multiplication, applied to the arguments.

Multilinearity as the Defining Behavior

Linearity in Each Argument Separately

The coordinate free description requires that fixing all arguments except one produces a linear function of the remaining argument, meaning the tensor's output scales and adds correctly with respect to that argument while the others are held fixed.

T (\dots, a u + b w, \dots) = a T (\dots, u, \dots) + b T (\dots, w, \dots)

Applies Uniformly to Every Slot

This linearity condition applies identically to every one of the tensor's slots, whether that slot accepts a vector or a covector, giving a single uniform rule that fully specifies the tensor's behavior across its entire domain.

Relating the Map Description to Components

Components as a Consequence, Not a Starting Point

Once a basis is introduced, the coordinate free map description determines the tensor's components automatically, by evaluating the map on the basis vectors and dual basis covectors. The components are thus a derived consequence of the map description, rather than an independent definition.

Recovering the Map from Components

Conversely, given a complete component array and a basis, the original map can be reconstructed by using multilinearity to extend the action on basis elements to an action on arbitrary vectors and covectors expressed in that basis.

Advantages of the Map Description

Immediate Basis Independence

Because the description never invokes a basis, every property derived directly from it, such as tensor type, multilinearity, and behavior under composition with other maps, is automatically basis independent, without requiring any separate verification.

Clarity About What a Tensor Fundamentally Is

The map description makes explicit that a tensor is, at its core, a rule for producing scalars from vector and covector inputs, which clarifies why component arrays, useful as they are for computation, are only one particular way of recording that underlying rule.