Activity: Vector Differential--Curvilinear

Vector Calculus II 2022 (8 years)

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

  • Media
    • activity_media/vfdrvectorcurvihand.pdf
    • activity_media/cylindrical_dr.png
    • activity_media/spherical_dr.png
    • activity_media/cylinder20_cs2LU37.png
    • activity_media/sphere20_6QRiBiW.png

Cylindrical Coordinates:

Find the general form for \(d\vec{r}\) in cylindrical coordinates by determining \(d\vec{r}\) along the specific paths below.

  • Path 1 from \((s,\phi,z)\) to \((s+ds,\phi,z)\): \[d\vec{r}=\hspace{35em}\]
  • Path 2 from \((s,\phi,z)\) to \((s,\phi+d\phi,z)\): \[d\vec{r}=\hspace{35em}\]
  • Path 3 from \((s,\phi,z)\) to \((s,\phi,z+dz)\): \[d\vec{r}=\hspace{35em}\]

If all three coordinates are allowed to change simultaneously, by an infinitesimal amount, we could write this \(d\vec{r}\) for any path as:

\[d\vec{r}=\hspace{35em}\]

This is the general line element in cylindrical coordinates.

Figure 1: \(d\vec{r}\) in cylindrical coordinates


Spherical Coordinates:

Find the general form for \(d\vec{r}\) in spherical coordinates by determining \(d\vec{r}\) along the specific paths below.

  • Path 1 from \((r,\theta,\phi)\) to \((r+dr,\theta,\phi)\): \[d\vec{r}=\hspace{35em}\]
  • Path 2 from \((r,\theta,\phi)\) to \((r,\theta+d\theta,\phi)\): \[d\vec{r}=\hspace{35em}\]
  • Path 3 from \((r,\theta,\phi)\) to \((r,\theta,\phi+d\phi)\): (Be careful, this is a tricky one!) \[d\vec{r}=\hspace{35em}\]

If all three coordinates are allowed to change simultaneously, by an infinitesimal amount, we could write this \(d\vec{r}\) for any path as:

\[d\vec{r}=\hspace{35em}\]

This is the general line element in spherical coordinates.

Figure 2: \(d\vec{r}\) in spherical coordinates

Instructor's Guide

Main Ideas

This activity allows students to derive formulas for \(d\vec{r}\) in cylindrical, and spherical coordinates, using purely geometric reasoning. These formulas form the basis of our unified view of all of vector calculus, so this activity is essential. For more information on this unified view, see our publications, especially: Using differentials to bridge the vector calculus gap

Students' Task

Using a picture as a guide, students write down an algebraic expression for the vector differential in 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.

It may help to do activity Vector Differential--Rectangular as an introduction.

Student Conversations

For the case of cylindrical coordinates, students who are pattern-matching will write \(d\vec{r} = dr\, \hat{r} + d\phi\, \hat{\phi} + dz\, \hat{z}\). Point out that \(\phi\) is dimensionless and that path two is an arc with arclength \(r\, d\phi\).

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 \(d\vec{r} = dr\, \hat{r} + d\phi\, \hat{\phi} + d\theta\, \hat{\theta}\). It helps to draw a picture in cross-section so that they can see that the circle whose arclength gives the coefficient of \(\hat{\theta}\) 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 \(d\vec{r}\).


Author Information
Corinne Manogue, Tevian Dray, & Katherine Meyer
Learning Outcomes