Calculus for the Modern Engineer: Putting the Joy Back in Learning Advanced Mathematics

TL;DR

The paper addresses the misalignment between traditional calculus curricula and the computational needs of modern engineering. It presents a pilot where a 4-credit course reorders and unifies calculus topics, emphasizes computation in Julia, and ties theory to robotics-focused case studies via three pillar projects and an open-source textbook. The study reports strong student engagement and learning gains, with positive assessments and actionable feedback for future iterations, including broader applications and increased coding challenges. This work offers a scalable model for updating engineering mathematics education to integrate computation, visualization, and real-world problem solving.

Abstract

Many engineering students enter college excited about math and physics, only to have their enthusiasm dimmed by a rigid, outdated calculus curriculum. The University of Michigan's Robotics Department is piloting a new 4-credit course, ``Calculus for the Modern Engineer,'' to reintroduce the excitement of learning advanced mathematics. Integrating Differential and Integral Calculus, vector derivatives, and Ordinary Differential Equations (ODEs) into a unified one-semester curriculum, the course emphasizes conceptual mastery and real-world applications. It starts with definite integration -- building on students' intuitive understanding of sums -- before progressing through limits, differentiation, antiderivatives, and ODEs. By leveraging computational tools like Julia, Large Language Models (LLMs), and Wolfram Alpha Pro, it reduces reliance on tedious hand calculations. Case studies in numerical integration, optimization, and feedback control connect theory to engineering challenges. Supported by an open-source textbook and programming assignments, the course equips students with modern computational skills and reignites their passion for mathematics.