7.22.1 Tensor Component Indexed Symbol

Tensor Component Indexed Symbol is the written notation consisting of a base letter together with one or more attached indices, placed either as superscripts or subscripts, used to denote a single component of a tensor at a specific, though generally unspecified, combination of index values.

Structure of the Symbol

The Base Letter

The base letter of a Tensor Component Indexed Symbol identifies which tensor the symbol refers to, with different tensors conventionally assigned different base letters so that expressions involving several tensors at once remain unambiguous. The base letter alone, without any attached indices, refers to the tensor as a whole rather than to any particular component.

The Attached Indices

Each index attached to the base letter identifies one of the tensor's index positions and indicates, through its placement above or below the base letter, the variance type associated with that position. An index written as a superscript denotes a contravariant position, while an index written as a subscript denotes a covariant position, so that the symbol T with a superscript i and a subscript j, written as T superscript i subscript j, denotes a component of a mixed tensor with one contravariant and one covariant index.

Illustration

The base letter identifies the tensor, while each index, positioned as a superscript or subscript, identifies one index position of the tensor and its associated variance type.

Free and Fixed Indices Within a Symbol

Free Indices Denote an Entire Family of Components

When an index in a Tensor Component Indexed Symbol is left as a letter rather than replaced with a specific numerical value, it is called a free index, and the symbol as written refers not to a single number but to the entire family of components obtained by letting that index range over its full Tensor Component Index Range. A symbol with several free indices refers to the entire component table of the tensor across every combination of those indices.

Fixed Indices Denote a Single Numerical Component

When an index is instead replaced with a specific numerical value, the symbol refers to exactly one component of the tensor, corresponding to that particular choice of index value. Writing T superscript 1 subscript 2, for instance, denotes a single specific number, in contrast to the family of values denoted by T superscript i subscript j.

Consistency Requirements for the Symbol

Matching Index Letters Across an Expression

Within a single expression involving several indexed symbols, the same letter used as an index in more than one place is understood to refer to the same value of that index throughout the expression, unless the context of a summation convention specifies otherwise. Consistent use of index letters is essential to ensuring that an expression built from several Tensor Component Indexed Symbols has an unambiguous meaning.

Order of Indices Matters

The left-to-right order in which indices are attached to a base letter is significant, since exchanging the order of two indices may correspond to an entirely different component unless the tensor is known to follow the Tensor Component Symmetric Equality Rule or the Tensor Component Sign Change Rule for that particular pair. The notation itself does not assume any such relationship unless it has been separately established.

Relationship to Other Tensor Concepts

Tensor Component Indexed Symbol provides the basic written unit from which every formula and rule within the broader Tensor Component Notation is built, including the expressions used to state the Tensor Component Symmetric Equality Rule, the Tensor Component Sign Change Rule, and the transformation law underlying Tensor Component Object Preservation.