7.22 Tensor Component Notation

Tensor Component Notation is the collected system of written conventions used to represent a tensor's components and the operations performed upon them, encompassing the use of indexed symbols, the placement of indices as superscripts or subscripts to denote variance type, and the summation convention used to abbreviate repeated sums over shared indices.

The Building Blocks of the Notation

Indexed Symbols

At the foundation of the notation lies the Tensor Component Indexed Symbol, consisting of a base letter identifying the tensor together with attached indices identifying its index positions and their variance type. Every other convention within Tensor Component Notation is built from this basic unit, combining several indexed symbols according to further rules to express more elaborate relationships.

Superscript and Subscript Placement

The notation distinguishes contravariant indices, written as superscripts, from covariant indices, written as subscripts, allowing a single glance at an indexed symbol to reveal both how many indices a tensor carries and how those indices are divided between the two variance types. This placement convention is essential to correctly applying the transformation law that underlies Tensor Component Object Preservation, since contravariant and covariant indices transform according to different rules.

The Summation Convention

Abbreviating Repeated Sums

A central feature of Tensor Component Notation is the summation convention, under which an index appearing exactly once as a superscript and once as a subscript within a single term is understood to be summed over its full Tensor Component Index Range, without an explicit summation symbol being written. Under this convention, an expression such as:

is understood to mean the full sum over every value of i from 1 to n, without needing to write a summation symbol explicitly.

Free Indices Versus Summed Indices

Within an expression governed by the summation convention, an index that appears exactly once, matching neither a superscript nor a subscript counterpart elsewhere in the same term, remains a free index, referring to an entire family of values as it ranges over its Tensor Component Index Range. An index that appears exactly twice, once as a superscript and once as a subscript, is summed and no longer appears as a free index in the result of the expression.

Illustration

The index j, appearing once as a subscript and once as a superscript, is summed according to the convention, while the index i, appearing only once, remains free and identifies a whole family of resulting values.

Additional Notational Devices

Parentheses and Brackets for Symmetrization

Tensor Component Notation includes special bracket conventions to denote symmetrized or antisymmetrized combinations of indices, with round parentheses enclosing indices to denote symmetrization according to the Tensor Component Symmetric Tensor Role, and square brackets enclosing indices to denote antisymmetrization according to the Tensor Component Exterior Tensor Role.

Comma and Semicolon Notation for Derivatives

In contexts involving derivatives of tensor components, the notation extends further by attaching a comma before an additional lower index to denote an ordinary partial derivative, and a semicolon before an additional lower index to denote a derivative that accounts properly for changes in the underlying coordinate system, preserving the tensorial nature of the result.

Purpose of a Unified Notation

Compactness Without Loss of Precision

Tensor Component Notation allows lengthy expressions involving many terms and summations to be written compactly, while still conveying the exact meaning of the expression through the precise placement and repetition of indices, avoiding the need to write out explicit summation symbols or lengthy verbal descriptions.

Directly Reflecting the Transformation Law

Because the notation distinguishes contravariant from covariant indices explicitly, expressions written in this notation directly reflect how each component would transform under a change of coordinates, reinforcing the connection between the written notation and Tensor Component Object Preservation.

Relationship to Other Tensor Concepts

Tensor Component Notation provides the written framework within which every concept discussed throughout the study of Tensor Components, including the Tensor Component Symmetric Equality Rule, the Tensor Component Sign Change Rule, and the various forms of Tensor Component Interpretation, is expressed, built up from the basic Tensor Component Indexed Symbol through the summation convention and its accompanying notational devices.