1.4 Tensor Description Foundations

Tensor Description Foundations is the body of concepts concerned with the various complementary ways a tensor can be presented, named, and notated, spanning the abstract coordinate-free description of a tensor as a multilinear map or an element of a tensor product space, the concrete coordinate description via an indexed array of components relative to a basis, and the several notational conventions, index notation, abstract index notation, and matrix notation, used to record and manipulate these descriptions. Where tensor definition foundations establish what a tensor is and tensor structure foundations establish how tensors relate to one another algebraically, tensor description foundations establish the vocabulary and notational apparatus used to communicate about tensors precisely.

Coordinate-Free Description

The Abstract Object

A tensor can be described entirely without reference to any basis, either as a multilinear map from copies of V and V* to the field F, or as an element of the abstractly constructed tensor product space T^p_q(V). This coordinate-free description emphasizes that the tensor exists as a single, well-defined mathematical object prior to any choice of coordinates.

Advantages of the Abstract Description

Describing a tensor abstractly makes basis-independent statements, such as identities and equations relating several tensors, immediately transparent, since no coordinate system need be introduced or tracked, and any relation established abstractly automatically holds in every basis once translated into components.

Coordinate Description via Components

The Indexed Array

Once a basis of V is fixed, a tensor can be described concretely by the array of its components, T^{i_1...i_p}_{j_1...j_q}, a collection of n^(p+q) scalars indexed by the chosen basis. This coordinate description is the form most directly amenable to explicit calculation, numerical computation, and comparison with other coordinate-dependent quantities.

Advantages of the Coordinate Description

Describing a tensor through its components allows direct arithmetic manipulation, explicit evaluation, and concrete representation as arrays or matrices suitable for computer implementation, at the cost of requiring the transformation law to be tracked whenever the basis changes.

Notational Conventions

Index Notation

Index notation, also called Ricci calculus, writes tensors with explicit upper and lower indices attached to a base symbol, combined with the Einstein summation convention that automatically sums over any index repeated once as an upper index and once as a lower index within a single term.

Abstract Index Notation

Abstract index notation retains the letters used in index notation, but the letters denote the type and slot structure of a tensor rather than specific numerical values, allowing an equation such as T^a_b to represent the tensor itself, valid in every basis simultaneously, rather than a specific numerical array tied to one coordinate system. This blends the clarity of index bookkeeping with the basis independence of the abstract description.

Matrix and Array Notation

For low-rank tensors, particularly type (1, 1) and type (0, 2) tensors, components are often displayed using ordinary matrix notation, arranging the entries into rows and columns, which connects tensor description directly to the tools and terminology of elementary linear algebra.

Passing Between Descriptions

From Abstract to Coordinate

Given the abstract multilinear map description of a tensor, its coordinate description in a chosen basis is obtained by evaluating the map on all combinations of basis vectors and dual basis covectors, producing the array of components directly.

From Coordinate to Abstract

Given a coordinate description, the abstract tensor is reconstructed by summing the components against the corresponding basis tensor products, recovering the same basis-independent object regardless of which basis was used to compute the components in the first place.

Consistency Across Descriptions

The two descriptions, abstract and coordinate, and the notational conventions built upon them, are guaranteed to agree precisely because the transformation law governing components is derived from, and consistent with, the underlying multilinear structure; no description offers information unavailable in the others, and the choice among them is a matter of convenience for the task at hand rather than a change in mathematical content.

Choosing a Description for a Given Purpose

Proof and Structural Argument

Abstract, coordinate-free descriptions are typically preferred when establishing general identities or structural properties of tensors, since such arguments avoid the bookkeeping overhead of tracking basis changes explicitly.

Computation and Explicit Evaluation

Coordinate descriptions using explicit components are typically preferred when a specific numerical answer is required, when implementing a tensor computation in software, or when comparing a tensor's behavior directly against a chosen reference basis.

Diagrammatic Summary

The diagram shows the two-way correspondence between the abstract, basis-free description of a tensor and its concrete coordinate description as an array of components relative to a chosen basis, with each description recoverable from the other.