12.20 Tensor Operation Notation

Tensor Operation Notation is the collection of symbolic conventions used to write down tensor operations unambiguously, encompassing index notation with raised and lowered indices, abstract coordinate-free symbols such as ⊗, f*, and f_*, the summation convention for repeated indices, and the typographical devices — subscripts, superscripts, stars, and bars — that distinguish operations acting on covariant slots from those acting on contravariant slots. Because a single tensor operation can be written in several outwardly different but logically equivalent ways, tensor operation notation exists to fix, for a given context, exactly which symbols mean what, so that an expression can be parsed correctly without ambiguity about which operation, which slots, and which spaces are intended.

Index Notation

Upper and Lower Indices for Variance

By nearly universal convention, superscript indices denote contravariant slots and subscript indices denote covariant slots, so a tensor of type (p,q) carries p upper indices and q lower indices. This placement convention is itself a piece of tensor operation notation: it encodes variance type directly into the symbol, allowing type compatibility to be checked by inspection of index position alone.

The Einstein Summation Convention

Under the summation convention, an index appearing once as a superscript and once as a subscript in the same term is automatically summed over, with the explicit summation symbol Σ suppressed entirely. This notational shorthand is what allows contraction to be written as simple juxtaposition of indices, TⁱSᵢ, rather than requiring an explicit sum sign every time a contraction occurs.

Coordinate-Free Symbolic Notation

The Tensor Product Symbol

The symbol ⊗ denotes the tensor product operation applied to two tensors (or vectors), producing a higher-rank tensor without reference to any coordinate system. This symbol is read consistently across contexts — vector-vector, tensor-tensor, or space-space (as in V ⊗ W) — with its precise meaning determined by what kind of objects appear on either side of it.

Star Notation for Pullback and Pushforward

A star placed as a superscript on the source map, f*, conventionally denotes the pullback operation induced by f, while a star placed as a subscript, f*, denotes the pushforward. This superscript-subscript distinction mirrors the upper-lower index convention for variance: the pullback acts on covariant (lower-index) objects and is marked with an upper star, while the pushforward acts on contravariant (upper-index) objects and is marked with a lower star.

Diagram of Notation Encoding Meaning

Notational Conventions for Composite and Special Operations

Parenthesized Slot Labels for Contraction

When a tensor has several slots of the same variance and a contraction must specify exactly which contravariant slot pairs with which covariant slot, a subscript pair such as C₁,₂ names the slot positions explicitly, resolving the slot compatibility ambiguity that plain index notation alone can leave unclear when more than one candidate pairing exists.

Bar and Prime Notation for Related but Distinct Bases

A bar, v̄, or a prime, v′, placed over or after a symbol conventionally indicates the same underlying vector or tensor expressed relative to a different, related basis, distinguishing coordinate expressions that refer to one mathematical object without introducing an entirely new variable name for what is conceptually the same tensor.

Consistency Requirements Across a Single Document or Computation

Notation Must Be Fixed Once and Used Uniformly

Because several of these conventions — index placement for variance, star placement for pullback versus pushforward — are conventions rather than logical necessities, a single computation or document must fix its notational choices at the outset and apply them consistently; switching conventions partway through (for instance, using lower indices for contravariant slots in one section and upper indices for the same purpose elsewhere) reintroduces exactly the ambiguity the notation exists to prevent.

Notation as a Compressed Encoding of Compatibility Requirements

Well-designed tensor operation notation is not merely decorative shorthand; the index-position and star-placement conventions directly encode the type compatibility and slot compatibility requirements discussed elsewhere, so that a well-formed expression in standard notation is, by construction, far more likely to already satisfy those compatibility requirements than an expression using ad hoc or inconsistent symbols.

Practical Value of Standardized Notation

Enabling Rapid Type Checking by Inspection

A reader familiar with the standard conventions can verify variance type compatibility, detect implied summations, and identify which specific operation (pullback, pushforward, contraction, tensor product) a given expression represents purely by reading the placement of indices and symbols, without needing to consult a separate, explicit statement of the operation's definition each time it appears.

Facilitating Translation Between Index and Coordinate-Free Styles

Because index notation and coordinate-free symbolic notation (such as ⊗, f*, f_*) describe the same underlying operations, a firm grasp of both notational systems allows a tensor expression to be translated between the two styles as needed, choosing whichever notation makes a particular argument or computation clearest in context.