12.20.3 Tensor Evaluation Notation

Tensor Evaluation Notation systematizes tensor component computation via indices, linking abstract algebra to applied physics and engineering calculations.

Tensor Evaluation Notation is the specific set of symbolic conventions used to write down the act of feeding a tuple of vectors, covectors, or mixed arguments into a tensor viewed as a multilinear map, encompassing functional-application notation such as T(v₁,...,vₖ), angle-bracket pairing notation such as ⟨T, v⟩, and index notation in which evaluation against basis elements is expressed by simply attaching matching indices. It is the specific corner of tensor operation notation concerned with the single most basic act underlying every other tensor operation — plugging concrete vectors into a tensor's argument slots — and fixing how that act is written is a prerequisite for writing down any operation, such as pullback, pushforward, or contraction, whose defining formula itself is stated in terms of evaluation.

Functional Application Notation

Direct Argument-List Notation

T (v_{1}, \dots, v_{k})

A covariant k-tensor T, viewed as a multilinear map, is evaluated on a tuple of vectors (v₁,...,vₖ) using ordinary function-application notation, exactly as an ordinary function f(x) is evaluated on an argument x. This notation makes the multilinear-map perspective on tensors explicit and is the notation used throughout the defining formulas of the pullback and pushforward operations.

Evaluation Order Corresponds to Slot Order

Because the arguments of T are listed in a fixed order corresponding to a fixed ordering of T's slots, the position of vᵢ in the argument list T(v₁,...,vₖ) must match the position of the i-th slot in whatever slot-ordering convention has been fixed for T, tying evaluation notation directly to the slot compatibility requirements discussed for slot-specific tensor operations.

Pairing Notation

Angle Brackets for Covector-Vector Pairing

⟨ ω, v ⟩ = ω (v)

For a covector ω (a rank-1 covariant tensor) paired against a vector v, the angle-bracket notation ⟨ω, v⟩ is often used interchangeably with the direct functional notation ω(v), emphasizing a symmetric, bilinear-pairing perspective on the evaluation rather than a one-sided function-application perspective. Both notations denote the identical scalar value; the choice between them is stylistic, reflecting whether the evaluation is being conceptually treated as "applying a function" or "pairing two objects."

Extension to Higher-Rank Pairings

⟨ T, v_{1} \otimes \dots \otimes v_{k} ⟩ = T (v_{1}, \dots, v_{k})

The angle-bracket pairing extends naturally to evaluate a covariant k-tensor T against a single simple contravariant k-tensor v₁⊗...⊗vₖ, treating the entire tuple of arguments as one combined object being paired against T. This extension is what allows the evaluation of a tensor to be written as a single pairing operation regardless of rank, rather than requiring a variable-length argument list notation for each different rank.

Index Notation for Evaluation on Basis Elements

Evaluation Against Basis Vectors Defines the Coordinates

T_{i_{1} \dots i_{k}} = T (e_{i_{1}}, \dots, e_{i_{k}})

The coordinates Ti₁...ik of a covariant tensor T relative to a basis {eᵢ} are, by definition, exactly the values obtained by evaluating T on every possible tuple of basis vectors. This equation is itself an instance of tensor evaluation notation and is the precise link between the functional-application view of a tensor and its more computational, coordinate-array representation.

Evaluation on a General Vector via Coordinates

T (v) = T_{i} v^{i}

For a general vector v = vⁱeᵢ (using the summation convention), evaluating a covector T on v is written using index notation as Tᵢvⁱ, with the implied summation over the repeated index i. This is exactly evaluation notation combined with summation-convention notation, showing how the two notational systems interact when a tensor is evaluated on an arbitrary, non-basis vector.

Diagram of the Three Evaluation Notations for the Same Value

Consistency Requirements for Evaluation Notation

Matching Argument Count to Tensor Rank

Regardless of which notational style is used, the number of arguments supplied in an evaluation must match the covariant rank of the tensor being evaluated; supplying k−1 or k+1 arguments to a rank-k covariant tensor produces an expression with no defined meaning, since the tensor's multilinearity is only guaranteed to be well-defined on tuples of exactly its own rank.

Consistent Choice of Notation Within a Single Derivation

Because functional notation, pairing notation, and index notation are freely interchangeable but visually distinct, a single derivation should generally commit to one style throughout, switching between them only at clearly marked points, so that a reader is never left uncertain whether a shift in notation reflects a shift in the underlying mathematical content or is purely stylistic.

Role of Evaluation Notation in Defining Other Operations

The Basis for Every Higher-Level Operation's Defining Formula

Every tensor operation whose definition involves substituting specific arguments — the pullback's slot transformation, the pushforward's vector slot action, contraction's summation over matched slots — is ultimately written using one of the evaluation notations described here. Fixing evaluation notation clearly is therefore a prerequisite for stating these higher-level operations' formulas without ambiguity, since those formulas are themselves built directly out of evaluation expressions.