In this post, I present a conceptual question that is so important, I feel that no undergraduate student should complete an introductory calculus-based physics course without being able to correctly answer it. Given the way introductory calculus and introductory physics are traditionally taught, I just don’t think there is nearly enough carryover from the calculus to the physics. This is especially true for setting the stage for vectors. Ironically, whether or not vectors were “covered” in students’ introductory calculus courses, students have in fact seen vectors all along, but they just didn’t call them that. They called them partial derivatives. With that in mind, here’s the question:
Discuss the relationship between the concepts of “conservative force” from introductory physics and “exact differential” from introductory calculus. Go get started, if you need a hint, consider an arbitrary function f(x,y,z) and write its total derivative.
My prediction is that most student will not be able to answer this question correctly. As always, I’d be interested in hearing from you and your students if you show them this question.