6.24.5 Tensor Order Notation Convention

Tensor Order Notation Convention is the set of accepted stylistic practices for referring to a tensor's order in prose and in writing when the contravariant-covariant split is not the point being made, covering the competing terms "order," "rank," and "degree," the way order is stated alongside or instead of the type pair, and the situations in which each choice of wording is preferred by different mathematical communities. This convention governs word choice and phrasing rather than symbolic notation, and it exists because several different terms have historically been used for the same underlying quantity, creating a need for explicit guidance on which term to use and when.

Competing Terms for the Same Quantity

Order

"Order" is the term favored when discussing tensors in the context of multilinear algebra broadly, and it is the term least likely to cause confusion with unrelated concepts, since "order" in other branches of mathematics (group order, order of an element) is understood from context to be a different notion when tensors are under discussion.

Rank

"Rank" is used in some texts as a direct synonym for order, but this usage risks collision with the well-established, unrelated notion of matrix rank (the dimension of a linear map's image); the order notation convention therefore recommends reserving "rank" for the matrix-theoretic meaning whenever both concepts might plausibly appear in the same discussion, and using "order" for the index count instead.

Degree

"Degree" appears occasionally, particularly in older or more algebraically oriented texts, as a synonym for order, echoing the language of polynomial degree since a tensor of order n behaves in certain respects like a degree-n homogeneous polynomial expression in coordinates; this term is less common than "order" in contemporary usage but appears often enough in the literature that recognizing it as a synonym prevents confusion when reading older sources.

Combining Order Terminology With Type

Order Stated Alongside Type

is a fully explicit phrasing that states both quantities even though order is redundant given the type (since order = p + q); this redundancy is intentional and aids readability, allowing a reader to register the tensor's overall size before parsing the finer type information.

Order Stated Alone When Type Is Implicit or Irrelevant

When a discussion concerns only purely covariant or purely contravariant tensors, stating the order alone — "a rank-2 covariant tensor," "an order-4 tensor" — is common and unambiguous, since the word "covariant" or the established context already fixes the type given the order, making explicit repetition of the type pair unnecessary.

Diagram of the Terminology Landscape

Practical Guidance for Consistent Usage

Preferring Order in Mixed Contexts

Whenever a piece of writing discusses both tensors and matrix rank in close proximity, the order notation convention recommends using "order" exclusively for the index count and reserving "rank" for the matrix-theoretic invariant, eliminating any ambiguity that would otherwise require the reader to infer meaning from context on every occurrence.

Stating Type Pair Explicitly in Formal Definitions

In formal definitions and theorem statements, where precision outweighs brevity, stating the full type pair (p, q) alongside or instead of a bare order term is the safer convention, since it removes any residual ambiguity about the contravariant-covariant split that a bare order term, by design, does not resolve.