9.18 Tensor Coordinate Free Tensor Description

Tensor Coordinate Free Tensor Description is the overall approach of defining, discussing, and manipulating tensors entirely in terms of their abstract properties and behavior, without ever introducing a basis, coordinates, or indexed components. It is the umbrella framework encompassing the map-based characterization of tensors and the bare, unindexed way of referring to them, unified by the shared principle of setting coordinates aside entirely.

Core Principle of the Description

Tensors as Objects, Not Arrays

At the heart of the coordinate free description is the view that a tensor is fundamentally an abstract multilinear object, existing independently of any coordinate system, with component arrays regarded as merely one convenient but secondary way of recording that object once a basis happens to be chosen.

Two Complementary Aspects

The description combines a functional aspect, in which a tensor is specified by how it acts on vector and covector arguments, with a referential aspect, in which the tensor is named and manipulated using bare symbols that stand for the object as a whole.

Elements Composing the Description

Defining Behavior Through Multilinearity

The coordinate free description specifies a tensor's type and behavior entirely through its multilinearity in each argument, a property stated using only vector space operations and requiring no coordinates to express.

Naming and Combining Without Indices

Building on this behavior, the description allows tensors to be named with plain symbols and combined through operations such as addition, tensor product, and contraction, all expressed as relationships among these bare symbols rather than among indexed component arrays.

Relationship to Component-Based Description

A Prior, More Fundamental Layer

The coordinate free description is logically prior to any component-based description: components are obtained from the coordinate free tensor only after a basis is introduced, whereas the coordinate free description itself requires no such introduction at any stage.

Guaranteeing Consistency of Components

Because the coordinate free description defines a tensor without reference to a basis, any component array derived from it in a particular basis is automatically consistent with the standard transformation law, since that law is nothing more than the requirement that different bases must yield descriptions of the same underlying coordinate free object.

Advantages of Working Coordinate Free

Results That Hold Universally

Statements and derivations carried out entirely within the coordinate free description apply automatically in every basis, since no particular basis was ever invoked in reaching them, sparing the need to separately verify basis independence after the fact.

Conceptual Clarity

Working coordinate free keeps attention focused on the actual mathematical content of a relationship among tensors, rather than on the bookkeeping of indices, summations, and transformation rules that become necessary only once explicit computation in a specific basis is required.

Practical Use of the Description

Preferred for Definitions and General Theory

Foundational definitions, general theorems, and structural relationships in tensor algebra are typically presented using the coordinate free description, reserving the introduction of a basis for the stage where concrete numerical results must actually be produced.

A Framework to Return To

Even when a calculation proceeds through components for practical reasons, the coordinate free description remains the framework against which the meaning and validity of the final result should be checked, ensuring that what has been computed genuinely reflects a property of the tensors themselves.