Calculus with Julia
Calculus with Julia
Author: John Verzani
Published: August 31, 2026
Preface
Welcome to Calculus with Julia, a comprehensive collection of instructional notes designed to guide students through the study of calculus utilizing the Julia programming language.
Julia was selected for this project because it is an open-source language featuring a syntax that is intuitive and highly effective for mathematical computation.
🛠️ Getting Started
Before diving into the material, please complete the following:
- Review “Getting started with Julia” to handle installation and environment customization.
- Read “Julia interfaces” to explore the various ways you can interact with the software.
The Pedagogical Approach
Since the mid-1990s, there has been a concerted effort to diversify how calculus is taught. These notes adhere to the "Harvard style rule of four," which suggests that mathematical discourse should integrate four distinct perspectives:
"The conversation should include a graphical, numerical, algebraic, and verbal component."
While these notes use Julia to emphasize the numerical and graphical (and occasionally algebraic) elements, they differ from traditional Computer Algebra System (CAS) approaches.
Julia vs. Computer Algebra Systems (CAS)
| Feature | CAS (Mathematica, Maple, Sage) | Julia Approach |
|---|---|---|
| Primary Focus | Symbolic/Algebraic manipulation | Numerical computing |
| Integration | Deeply integrated symbolic engines | Symbolic treatment added as a layer |
| User Experience | High-level informal syntax (e.g., WolframAlpha/Siri) | Natural programming syntax |
| Learning Goal | Rapid production of finished products | Conceptual mastery via technology |
For instance, a user might ask Siri, "What is the graph of ?" and receive an instant result. While CAS tools are excellent for modeling symbolic treatments, they can sometimes make manual exercises feel redundant unnecessary because the answer is so easily obtained.
Our objective is to leverage technology to grasp core concepts without becoming overwhelmed by the intricacies of the programming language itself.
Scope and Structure
The notes focus on a streamlined subset of computational concepts. While this is sufficient for solving most calculus problems, it is not a comprehensive course in computer science. (However, curious students are encouraged to explore Julia's vast capabilities independently!)
Within this framework, most calculus operations are simplified into a standard function call:
action(function, arguments...)
Example implementation logic:
# Conceptual representation of a calculus action in Julia
result = compute_derivative(f, x_value)
Organization of the material:
- Focused Pages: Content is divided into pages targeting specific concepts, similar to textbook sections.
- Active Learning: Every page concludes with "try-it-yourself" exercises.
- Diverse Influences: The content draws from various sources, including Wikipedia, Angenent (2012), and Strang (2019).
Technical Implementation & Contributions
To streamline the experience, these notes are paired with the CalculusWithJulia package, which automates common tasks and pre-loads essential libraries.
- Platform: These notes are built using Quarto.
- Format: While primarily web-based, a PDF version can be compiled by assembling the various parts.
- Collaboration: To suggest a new topic, fix a typo, or correct an error, please use the “Edit this page” link to contribute.
Special Acknowledgments: A huge thank you to
@fangliu-tjufor their meticulous proofreading and the numerous Pull Requests (PRs) that improved the accuracy of this work.
Running Julia
For most users, the easiest way to install Julia is via the juliaup utility. For further details, please refer to the installation notes found in the Overview of Julia Commands.
Document Metadata:
- Version:
0.27 - Date: August 31, 2026
- Source Code: GitHub - SigurdAngenent/WisconsinCalculus
- Reference: MIT OCW Calculus Online Textbook