# Symbolics.jl: Symbolic Algebra Inside Julia Workflows > Symbolics.jl gives Julia programmers symbolic variables, differentiation, substitution, simplification, and Jacobians within ordinary Julia code. It fits mathematical software that needs symbolic expressions before numerical evaluation. **JuliaSymbolics/Symbolics.jl** — Symbolic programming for the next generation of numerical software - Repository: https://github.com/JuliaSymbolics/Symbolics.jl - Website: https://docs.sciml.ai/Symbolics/stable/ - Stars: 1,540 · Forks: 192 - Language: Julia - License: NOASSERTION - Published: 2026-10-16 · Updated: 2026-10-16 · Language: en - Canonical page: https://hysenlabs.com/projects/juliasymbolics-symbolics-jl ## Why Symbolics.jl belongs in Julia numerical software Symbolics.jl addresses the gap between working entirely with numbers and manipulating mathematical expressions before evaluation. Its documented operations include differentiation, solving symbolic systems of equations, substitution, simplification, and Jacobian calculation. This makes it relevant to engineers, domain scientists, researchers, and Julia programmers who build mathematical software and need symbolic behavior inside an existing application rather than through a separate symbolic interface. The package is written in Julia and is directly extendable in the same language. That matters because its scope is not limited to a fixed command-line CAS. A Julia package can represent unknowns, generate derivatives of expressions, apply substitution rules, and then pass symbolic or numerical results into the rest of its code. The README demonstrates these operations on scalar expressions and matrices, although it does not claim that every numerical workflow benefits from symbolic construction. Symbolics.jl sits between lower-level and higher-level tooling. SymbolicUtils.jl is the rule-rewriting foundation, while ModelingToolkit.jl adds symbolic-numeric representations for ODEs, SDEs, and related models. For a user who wants symbolic algebra operations directly, the README directs them to Symbolics.jl rather than asking them to assemble a CAS from SymbolicUtils.jl. ## Symbolic terms, dispatch, and rewriting The documented architecture begins with SymbolicUtils.jl, described as the core rule-rewriting system. Symbolics.jl builds its symbolic algebra capabilities on that foundation and adds operations such as differentiation and solving symbolic systems of equations. This separation gives Symbolics.jl a complete package while preserving a lower-level building block for projects that need to define a CAS for a specific algebra. The repository citation describes the implementation in terms of an abstract term interface and dynamic multiple dispatch. It says this design lets behavior vary according to the domain and the term type. The citation also describes a generic action interface through which different data structures can be used without rewriting existing term rewriters. These are project-specific architectural claims, but the README does not provide enough implementation detail to explain the internal call path for a particular simplification operation. The term representation remains symbolic until a later transformation changes it. In the example, `z = t + t^2` is followed by a differential operation and `expand_derivatives(D(z))`, which returns `1 + 2t`. Other expressions pass through `simplify`, `substitute`, or both. Matrix operations use Julia broadcasting, so expressions and substitution dictionaries are applied element by element rather than requiring a separate matrix-specific syntax. ## Installing Symbolics.jl through Julia Pkg The README uses Julia's package manager for installation. Open the Julia REPL, load `Pkg`, and add the package with the following commands: ```bash julia> using Pkg julia> Pkg.add("Symbolics") ``` The package manager adds Symbolics.jl to the active Julia environment. Load it in a later or same session with `using Symbolics`. The README does not document a standalone executable installer, a required system service, or a Docker image, so none should be assumed. ## Creating variables and differentiating an expression Load Symbolics.jl after installation, declare symbolic variables with `@variables`, and create a `Differential` for the variable of interest. The first derivative remains a symbolic representation. Calling `expand_derivatives` expands that representation into an ordinary symbolic expression. ```julia julia> using Symbolics julia> @variables t x y julia> D = Differential(t) julia> z = t + t^2 julia> D(z) Differential(t)(t + t^2) julia> expand_derivatives(D(z)) 1 + 2t ``` The displayed output provides the expected first use case: construct an expression, differentiate it with respect to `t`, then expand the derivative. The README does not document network, registry, or compatibility troubleshooting for installation failures. ## Computing Jacobians and simplifying matrices Symbolics.jl can calculate a Jacobian from a vector of expressions and a vector of differentiation variables. In the README, the system is nonlinear because `x + x*y` includes a product of both variables. The result is a `2×2 Matrix{Num}` whose entries remain symbolic. The example also constructs a symbolic matrix and applies `simplify` through broadcasting. Entries such as `t^2 + t + t^2` are reduced to `t + 2(t^2)`, while terms such as `2t + 4t` become `6t`. This shows that simplification is an explicit transformation in the workflow, not a requirement to replace symbols with numbers first. ```julia julia> Symbolics.jacobian([x + x*y, x^2 + y],[x, y]) 2×2 Matrix{Num}: 1 + y x 2x 1 julia> B = simplify.([t^2 + t + t^2 2t + 4t x + y + y + 2t x^2 - x^2 + y^2]) 2×2 Matrix{Num}: t + 2(t^2) 6t x + 2t + 2y y^2 ``` The Jacobian output contains symbolic variables, while the simplification example focuses on algebraic cleanup. The README does not specify when automatic simplification occurs, so callers should not assume that every operation canonicalizes expressions. ## Substituting symbols with expressions or numbers `substitute` applies mappings from symbolic variables to either new expressions or concrete numeric values. The README first substitutes `x` with `y^2`, then simplifies the resulting matrix. That sequence separates algebraic transformation from cleanup, which can make the intermediate symbolic result easier to inspect. Substitution also works when the replacements are floating-point values. The example replaces `x`, `y`, and `t` with `2.0`, `3.0`, and `4.0`, producing numerical matrix entries. Because the README uses broadcasting with a tuple containing one dictionary, this is a matrix-wide operation rather than a manually repeated substitution for each cell. ```julia julia> simplify.(substitute.(B, (Dict(x => y^2),))) 2×2 Matrix{Num}: t + 2(t^2) 6t 2t + y^2 + 2y y^2 julia> substitute.(B, (Dict(x => 2.0, y => 3.0, t => 4.0),)) 2×2 Matrix{Num}: 36.0 24.0 16.0 9.0 ``` The first result remains symbolic because `y` and `t` remain present. The second result is numeric because every declared variable has a replacement. The README does not discuss substitution failure diagnostics, algebraic assumptions, branch handling, or undefined expressions. ## Choosing Symbolics.jl over SymbolicUtils.jl or ModelingToolkit.jl SymbolicUtils.jl is the closest architectural alternative documented by the project, but it solves a different layer of the problem. It is a rule-rewriting system intended as the bare foundation for building a new CAS around a specific algebra. Symbolics.jl builds on it and exposes a complete symbolic algebra system with differentiation and symbolic equation solving. The boundary is implementation responsibility: use SymbolicUtils.jl when the objective is to construct a specialized CAS, and use Symbolics.jl when those algebra capabilities are needed as a package. ModelingToolkit.jl is a higher-level alternative for modeling and simulation. It uses Symbolics.jl for symbolic equations and operations such as differentiation, then adds common representations for ODEs, SDEs, and other model classes. It is not a like-for-like substitute for Symbolics.jl. ModelingToolkit.jl shifts the center of work toward model representation and SciML integration, while Symbolics.jl keeps the documented examples focused on general symbolic expressions, matrices, substitution, and derivatives. Symbolics.jl also has limits as a product choice. The README does not document interactive notebooks, plotting, a desktop interface, deployment targets, or a standalone executable. Users coming from a dedicated CAS should verify that the required operations and workflows exist before relying on it. ## Maintenance, releases, and license review The repository is not archived, and its last push was on 2026-10-16. The recorded releases include `v7.45.0` on 2026-10-08, `v7.44.1` on 2026-10-06, and `v7.44.0` on 2026-10-06. These facts show recent repository and release activity, but the README does not include a compatibility matrix for Julia versions or a long-term support policy. The repository identifies its license as a custom license that GitHub cannot classify, with terms in `LICENSE.md`. That prevents a safe conclusion about redistribution, modification, packaging, or organizational use from the label alone. This article does not interpret the license or offer legal advice. Anyone planning to incorporate Symbolics.jl into a distributed or closed-source product must inspect `LICENSE.md` for the actual permissions and conditions. The project asks users to cite an associated paper if they use it. It also links stable and development documentation, with development documentation containing unreleased features. For repeatable application code, the distinction matters: the stable documentation is the safer starting point for documented behavior, while development documentation can describe code that has not yet appeared in a release. ## Conclusion Adopt Symbolics.jl when symbolic expressions must integrate directly with a Julia codebase, especially for differentiation, substitution, simplification, or equation-related operations. Do not choose it merely as a general CAS or as the lowest-level foundation for a custom algebra system. Verify first that your required operations are documented in the stable documentation, that symbolic evaluation is acceptable for the relevant expressions, and that the terms in LICENSE.md match your intended use and distribution model. ## FAQ ### How do I install Symbolics.jl? Open Julia, load `Pkg`, and run `Pkg.add("Symbolics")`. This is the installation method given in the README. ### How do I differentiate an expression with Symbolics.jl? Declare variables with `@variables`, create a `Differential`, apply it to an expression, and use `expand_derivatives` when an expanded result is required. The README derives `1 + 2t` from `t + t^2` with respect to `t`. ### When should I use SymbolicUtils.jl instead of Symbolics.jl? The README describes SymbolicUtils.jl as the barebones rule-rewriting foundation for building a new CAS for a specific algebra. It directs users who need a full symbolic algebra system, including differentiation and symbolic equation solving, to Symbolics.jl. ### Does Symbolics.jl replace ModelingToolkit.jl? No. ModelingToolkit.jl uses Symbolics.jl and adds symbolic-numeric modeling features, including representations for ODEs and SDEs. Symbolics.jl provides the underlying symbolic algebra capabilities directly. ## Sources - [Issues](https://github.com/JuliaSymbolics/Symbolics.jl/issues) - [JuliaSymbolics/Symbolics.jl on GitHub](https://github.com/JuliaSymbolics/Symbolics.jl) - [Project website](https://docs.sciml.ai/Symbolics/stable/) - [README](https://github.com/JuliaSymbolics/Symbolics.jl/blob/master/README.md) - [Releases](https://github.com/JuliaSymbolics/Symbolics.jl/releases) --- Hysen Labs editorial analysis, written from the project's own repository and release notes. Cite the canonical page: https://hysenlabs.com/projects/juliasymbolics-symbolics-jl