An OpenMath Content Dictionary for Tensor Concepts

We introduce a new OpenMath content dictionary, named tensor1, containing symbols for the expression of tensor formulas. These symbols support the expression of non-Cartesian coordinates and invariant, multilinear expressions in the context of coordinate transformations. While current OpenMath symbols support the expression of linear algebra formulas using matrices and vectors, we find that there is an underlying assumption of Cartesian, or standard, coordinates that makes the expression of general tensor formulas difficult, if not impossible. In introducing these new OpenMath symbols for the expression of tensor formulas, we attempt to maintain, as much as possible, consistency with prior OpenMath symbol definitions for linear algebra.

💡 Research Summary

The paper addresses a fundamental limitation of the current OpenMath standard: its existing content dictionaries (linalg1‑linalg6) are designed around Cartesian (orthogonal) coordinate systems and treat vectors and matrices essentially as row‑vectors and column‑vectors related by a simple transpose. This design implicitly assumes a fixed Euclidean metric and a standard basis, which suffices for linear‑algebraic expressions but fails to capture the richer structure of tensors, especially when non‑Cartesian coordinates, curved manifolds, or general metric tensors are involved.

To overcome this gap, the authors introduce a new OpenMath content dictionary called tensor1. The dictionary defines a suite of symbols that make explicit the notions of tuples, indexed components, basis selection, and the distinction between covariant (lower) and contravariant (upper) indices. The key symbols are:

• tuple – constructs an ordered n‑tuple from its arguments.
• tuple‑selector – extracts the i‑th element of a tuple, allowing coordinates to be treated as an n‑tuple rather than a vector.
• Cartesian(i) – returns the i‑th Cartesian coordinate value.
• unitCartesian(i) – returns the i‑th unit basis vector of a right‑handed Cartesian frame.
• Kronecker‑tensor – the δᵢⱼ symbol for identity mapping of indices.
• basis‑selector – a binary operator that, given a basis tuple and an index (either a contra‑index or a covar‑index), returns the corresponding basis vector or basis covector.
• tensor‑selector – extracts a scalar component from a tensor; it takes three arguments: the tensor object, a tuple of index symbols (mix of contra‑ and covar‑indices), and a frame (ordered basis). The ordering of indices follows a row‑major convention, ensuring unambiguous storage mapping.
• contra‑index and covar‑index – symbols that denote upper (contravariant) and lower (covariant) indices respectively, making the transformation behavior of each index explicit.
• metric‑tensor – a symmetric, non‑degenerate tensor gᵢⱼ that defines the inner product and length element ds² = gᵢⱼ dx′ᵢ dx′ⱼ, thus generalizing the Euclidean metric.
• Levi‑Civita – the totally antisymmetric permutation tensor ε, parameterized by the dimension of the space, useful for expressing cross products and curl operations in component form.

The paper first reviews the mathematical background: a point in ℝⁿ is described by an n‑tuple of Cartesian coordinates (x₁,…,xₙ) and a position vector r = Σ xᵢ eᵢ, where eᵢ are fixed orthonormal basis vectors. Non‑Cartesian coordinates (x′ᵢ) are introduced as functions of the Cartesian ones, possibly nonlinear. The authors stress that while the tuple of coordinates resembles an OpenMath context (an n‑tuple of variables), it must not be treated as a vector because vector addition of coordinate tuples is not invariant under general coordinate transformations.

Vectors are expressed as v = Σ vᵢ gᵢ, where gᵢ are the (generally coordinate‑dependent) basis vectors, and covectors as u = Σ uᵢ gⁱ, with gⁱ(gⱼ)=δⁱⱼ. The transformation rules are explicitly given:
g′ᵢ = Σ ∂xⱼ/∂x′ᵢ gⱼ (covariant transformation of basis vectors),
v′ˡ = Σ ∂x′ˡ/∂xᵏ vᵏ (contravariant transformation of components), and
u′ⱼ = Σ ∂xᵏ/∂x′ⱼ uᵏ (covariant transformation of covector components).

Higher‑order tensors are built from outer products of vectors and covectors, followed by possible contractions. The order of a tensor equals the total number of vector and covector factors. To encode tensor components in markup, the tensor‑selector uses a tuple of index symbols; the order of the indices determines the layout in memory (row‑major). The authors argue that the Einstein summation convention, while common in handwritten mathematics, must be made explicit in OpenMath because there is no intrinsic notion of “bound index” in the markup language.

The dictionary also provides the Kronecker delta for identity mappings, the metric tensor for defining lengths and angles in arbitrary geometries, and the Levi‑Civita symbol for antisymmetric operations. By combining tuple, Cartesian, unitCartesian, and basis‑selector, one can construct arbitrary frames, define coordinate transformations, and then use tensor‑selector, contra‑index, covar‑index, metric‑tensor, and Levi‑Civita to write full tensor equations that are independent of the chosen frame.

In the conclusion, the authors list the twelve symbols introduced and illustrate how they enable the construction of Cartesian frames, the definition of coordinate transformations, the creation of tensor spaces on those frames, the assignment of component values, and the expression of tensor equations—including those involving non‑Euclidean metrics and curl operations. The dictionary is submitted to the OpenMath repository, and the authors acknowledge contributions from collaborators.

Overall, the paper makes a compelling case that extending OpenMath with a dedicated tensor content dictionary fills a critical gap for scientific and engineering communities that need machine‑readable, coordinate‑independent representations of tensor equations. The proposed symbols are designed to be compatible with existing OpenMath linear‑algebra symbols while providing the necessary semantics for covariant/contravariant indexing, basis selection, and metric‑dependent operations, thereby paving the way for broader adoption of OpenMath in fields such as continuum mechanics, general relativity, and advanced computational physics.