10.22 Tensor Change of Basis Notation

Tensor Change of Basis Notation is the collected set of symbolic conventions, covering index placement, coordinate labeling, summation, and matrix representation, that together make it possible to write tensor change-of-basis formulas unambiguously and compactly, forming the shared notational vocabulary underlying every specific transformation rule, check, and interpretation discussed elsewhere in the subject.

Coordinate Labeling Conventions

Bars, Primes, and Alternate Letters

Three common conventions distinguish target-chart quantities from source-chart quantities: an overline as in $${\overline{x}}^j$$, a prime as in $$x^{\prime j}$$, or an entirely different letter such as $$y^j$$ paired with $$x^i$$, with the choice among these three being a matter of typographical preference rather than mathematical substance, since all three consistently mark exactly the same distinction between source and target coordinates.

Numbering Versus Letter Indices

Coordinates and components may be indexed either with running numerical superscripts and subscripts, such as $$x^1,x^2,\dots$$, or with abstract letter indices such as $$x^i$$ standing for an arbitrary, unspecified coordinate direction, with letter indices used whenever a formula is meant to apply generally across every coordinate direction at once, and numerical indices reserved for concrete illustrative examples with a fixed, small number of dimensions.

Index Placement Conventions

Upper Versus Lower Position

An upper index marks a contravariant-type quantity and a lower index marks a covariant-type quantity, a distinction carried consistently across coordinates, basis vectors, dual basis vectors, and tensor components alike:

with basis vectors carrying a lower index despite being associated with contravariant transformation behavior, and dual basis vectors carrying an upper index despite being associated with covariant transformation behavior, an apparent inversion that is itself a standard and deliberate part of the notation, reflecting how each object pairs against the other type under the defining duality relation.

Jacobian and Inverse Jacobian Index Placement

The forward Jacobian is written with its target-chart index upper and its source-chart index lower, while the inverse Jacobian reverses this assignment, with the explicit minus-one superscript marking it as the inverse rather than serving as a tensor index itself:

Summation Notation Conventions

Explicit Summation Versus Einstein Convention

A transformation formula may be written with an explicit summation sign, as in $$\sum _i^n{J}_i^j{V}^i$$, or, more commonly in advanced treatments, with the summation sign suppressed entirely under the Einstein summation convention, relying solely on the repetition of the index $$i$$, once upper and once lower, to signal an implied sum.

Range of Summation

The upper limit of a written summation sign, typically denoted $$n$$, indicates the dimension of the space, and this same value applies uniformly to every summed index appearing throughout a given transformation formula unless explicitly stated otherwise.

Matrix Notation Conventions

Index Notation Versus Matrix Notation

The same transformation content can be written either in explicit index notation or in compact matrix notation, with the two related directly by treating the Jacobian as an ordinary matrix and the tensor components as a column or row array:

with matrix notation favored for compactness in low-rank cases and index notation favored whenever the specific pattern of upper and lower indices needs to be tracked explicitly, particularly for tensors of rank two or higher.

Diagram of the Notational Building Blocks

How the Pieces Fit Together

Notational Consistency Across the Subject

One Vocabulary for Every Topic

Every specific transformation rule, whether for a contravariant vector, a covariant vector, or a general mixed tensor, and every associated verification check, from index balance to component reconstruction, is written using this same shared set of notational conventions, so mastering these conventions once is sufficient to read and write any of the more specific formulas encountered throughout the study of tensor change of basis without needing to relearn notation from topic to topic.