Main Ideas

This activity allows students to discover formulas for \(d\ell\) in cylindrical, and spherical coordinates, using purely geometric reasoning.

Students' Task

Using a picture as a guide, students write down an algebraic expression for infinitesimal lengths in two different coordinate systems (cylindrical, spherical).

Introduction

Begin by drawing a curve (like a particle trajectory, but avoid "time" in the language) and an origin on the board. Show the position vector \(\vec{r}\) that points from the origin to a point on the curve and the position vector \(\vec{r}+d\vec{r}\) to a nearby point. Show the vector \(d\vec{r}\) and explain that it is tangent to the curve.

Student Conversations

For the case of cylindrical coordinates, students who are pattern-matching will write \(d\ell = d\phi\) on path 3. Point out that \(\phi\) is dimensionless and that path three is an arc with arclength \(s\, d\phi\).

Some students will remember the formula for arclength, but many will not. The following sequence of prompts can be helpful.

• What is the circumference of a circle?
• What is the arclength for a half circle?
• What is the arclength for the angle \(\pi\over 2\)?
• What is the arclength for the angle \(\phi\)?
• What is the arclength for the angle \(d\phi\)?

For the spherical case, students who are pattern matching will now write \(d\ell = r\, d\phi\). It helps to draw a picture in cross-section so that they can see that the circle whose arclength gives the coefficient of \(d\phi\) has radius \(r\sin\theta\). It can also help to carry around a basketball to write on to talk about the three dimensional geometry of this problem.

Wrap-up

The only wrap-up needed is to make sure that all students have (and understand the geometry of!) the correct formulas for \(d\ell\).

Extensions