3.18 Tensor Covector Notation

Tensor Covector Notation is the collection of symbolic conventions used to write, index, and manipulate elements of a dual vector space in a way that distinguishes them clearly from vectors and makes their transformation behavior and pairing rules transparent. Standard notation denotes a covector by a lower-case Greek letter such as ω or η, writes its components with a subscripted lower index ωᵢ, and expands it in a dual basis {eⁱ} of raised-index basis covectors as ω = ωᵢeⁱ, with summation over the repeated index implied by the Einstein convention. This notation is deliberately designed so that legality of an expression, meaning whether it can represent a basis-independent quantity, can often be checked by inspecting the placement of indices alone.

Core Symbolic Conventions

Lower Indices for Covariant Components

Covector components are written with the index as a subscript, as in ωᵢ, to mark them as covariant quantities, meaning they transform in the same direction as a change of basis on the underlying vector space rather than its inverse. This is paired with the convention of writing vector components with a superscript, vʲ, so that a mismatch in index height in a formula is an immediate signal that a proposed identity is not basis-independent.

The Dual Basis and Raised-Index Basis Covectors

The dual basis vectors themselves are written with a raised index, eⁱ, satisfying the biorthogonality relation with the basis of V,

where δⁱⱼ is the Kronecker delta. This notation ensures that when eⁱ is combined with the lower-index component ωᵢ in the expansion ω = ωᵢeⁱ, the repeated index appears once up and once down, matching the Einstein summation pattern and confirming that the sum represents a legitimate, basis-independent covector.

Einstein Summation and Index Contraction

The Summation Convention

Under the Einstein convention, any index that appears exactly twice in a single term, once as a subscript and once as a superscript, is automatically summed over its full range without an explicit summation sign. This convention was adopted specifically because covector-vector pairings, changes of basis, and tensor contractions overwhelmingly involve sums of exactly this shape, so suppressing the summation symbol reduces notational clutter without any loss of precision.

Free Indices Versus Dummy Indices

In covector notation, an index that is repeated and summed is called a dummy or contracted index and may be renamed freely, such as ωᵢvⁱ = ωⱼvʲ, while an index that appears only once in every term of an equation is called a free index and must match in name and height on both sides of any valid tensorial equation. This distinction is essential for correctly parsing expressions involving several covectors or vectors simultaneously.

Notation for the Pairing and for Pullback

Angle Bracket and Direct Application Notation

The evaluation of a covector on a vector is written either as direct function application, ω(v), or using angle bracket pairing notation, ⟨ω, v⟩, both denoting the same scalar Σᵢ ωᵢvⁱ. The angle bracket form emphasizes the bilinear, pairing-like character of the operation and is especially common when V* and V are being treated symmetrically as dual objects rather than one being viewed as functions on the other.

Star Notation for Pullback

The pullback of a covector ω on W under a linear or smooth map T: V → W is denoted T*ω, with the asterisk indicating the contravariant, index-reversing direction of the induced map on covectors relative to the direction of T itself on points or vectors. In components, if T has matrix representation Tⁱⱼ relative to chosen bases, the pullback is written

where the free index j on the right matches the free index j on the left, and the dummy index i is contracted, exhibiting the notation's self-consistency check directly in the formula.

Notation for Higher-Rank Covariant Objects

Multiple Lower Indices

A covariant tensor of rank two, such as a bilinear form or metric, is written with two lower indices, gᵢⱼ, and a general covariant tensor of rank k carries k lower indices, Tᵢ₁ᵢ₂...ᵢₖ. Symmetric tensors satisfy Tᵢⱼ = Tⱼᵢ, while antisymmetric tensors, such as those underlying differential forms, satisfy Tᵢⱼ = −Tⱼᵢ, and notation for the latter is often further abbreviated using wedge products of basis covectors, eⁱ ∧ eʲ.

Mixed-Index Tensors

Tensors combining both covariant and contravariant slots carry indices in both positions, as in Tⁱⱼ for a linear map viewed as a (1,1) tensor, and the notation prescribes that contraction between a mixed tensor and a covector or vector always pairs a lower index of one factor with a matching upper index of the other, preserving the same up-down contraction rule used for simple covector-vector pairing.

Notational Variants Across Contexts

Row Vector and Matrix Notation as a Parallel System

Alongside index notation, covectors are frequently written as row matrices, with the pairing ω(v) realized as literal matrix multiplication ωv of a 1×n row by an n×1 column. This matrix notation and the index notation ωᵢvⁱ describe the identical operation, and translating between them consistently, row form corresponding to lower-index quantities and column form to upper-index quantities, is a standard notational skill in working with dual spaces.

Coordinate-Free Notation

In more abstract treatments, covectors are denoted without any reference to components at all, using expressions such as ω ∈ V* or df for the differential of a function f, emphasizing that the covector notation ultimately serves a coordinate-independent object, with indexed and row-vector notations serving as computational tools for representing that object once a basis has been fixed.