Introduction
When we think of mathematical equations, the first thing that comes to mind is the familiar linearalgebraic or calculus notation taught in school. However, mathematics is a living language, and many fields have developed alternative ways of writing equations that suit their specific needs. These nonstandard formats might look unusual at first glance, but they often reveal structural information, improve readability, or enable automated processing.
This page gives an overview of why nonstandard equation formats exist, lists the most frequently encountered types, and offers practical suggestions for creating, reading, and converting them.
Why Use NonStandard Forms?
- Domainspecific clarity: In physics, chemistry, or finance, symbols or layout conventions capture concepts that would be cumbersome in plain algebraic form.
- Compactness: Large systemssuch as those arising in network theorycan be expressed in matrix or tensor notation, dramatically reducing visual clutter.
- Computational readiness: Many computeralgebra systems require a particular syntax to parse equations correctly.
- Historical tradition: Certain disciplines preserve older notations (e.g., differential forms in differential geometry) for continuity with classic literature.
Common NonStandard Formats
1. Matrix & Vector Notation
Instead of writing a system of linear equations one by one, we often bundle the coefficients into a matrix A and the unknowns into a column vector x, yielding the compact form Ax = b. This format highlights linearalgebraic properties such as rank, eigenvalues, and invertibility.
2. Tensor Index Notation
In continuum mechanics and relativity, tensors are expressed using superscripts and subscripts that obey Einsteins summation convention. For instance, the stressstrain relationship may be written as = C , where repeated indices imply summation over the dimension.
3. Functional Notation with Arrows
Category theory and functional programming use arrows to denote mappings: f : A B. When equations involve multiple morphisms, diagrams become the preferred medium, but a linear representation can still be written as g f = h.
4. Differential Forms
Rather than writing ddx (x) = 2x, the exterior derivative notation expresses differentiation on manifolds: d(x) = 2xdx. The wedge product replaces ordinary multiplication for oriented area elements.
5. LaTeXlike Inline Syntax
In wikis and markup languages, equations often appear inside delimiters such as \( ... \) or $$ ... $$. Though not a visual style, this encoding is a nonstandard textual format that many parsers convert to proper math fonts.
Tools & Tips for Working with NonStandard Equations
1. Use a MathFriendly Editor
Editors like Overleaf, MathType, or Visual Studio Code with the LaTeXWorkshop extension render LaTeX syntax in real time, improving the feedback loop when you experiment with unusual notations.
2. Convert Between Formats
Tools such as pandoc can translate Markdown with LaTeX math into HTML, DOCX, or PDF. For matrixoriented work, libraries like SymPy or NumPy can output LaTeX strings directly from Python objects.
3. Validate Syntax
When writing custom notations (e.g., domainspecific DSLs), simple regular expressions can catch common errors before the document reaches a parser. For tensor indices, ensure that each superscript matches a subscript where summation is intended.
4. Keep a Notation Glossary
Because nonstandard formats can be opaque to newcomers, a short table explaining symbols, index conventions, and any shorthand used in the document saves readers time.
Illustrative Examples
Example 1 Linear System in Matrix Form
2 -1 0 x 1-1 2 -1 x = 0 0 -1 2 x 3
Written compactly as Ax = b with A = [[2,-1,0],[-1,2,-1],[0,-1,2]], x = (x,x,x), b = (1,0,3).
Example 2 Tensor Contraction
Suppose C is a fourthorder elasticity tensor and is the strain tensor. The stress components are obtained by contracting over the middle indices:
= C
In Einstein notation the summation symbols are omitted, giving the concise representation shown earlier.
Example 3 Differential Form Identity
The exterior derivative of a 1form = fdx satisfies:
d = df dx = (f/ydy + f/zdz) dx
Because the wedge product is antisymmetric, the term dx dx vanishes, leaving only the crossterms.
Example 4 Functional Composition in Arrow Notation
Given f : , f(x) = x and g : , g(y) = e, their composition is expressed as:
g f : , (g f)(x) = e
In a linear text, one might write gf = h where h(x) = e^{x}.
