Small Group Activity: Vector Differential--Curvilinear
Static Fields 2023
group Small Group Activity
schedule
30 min.
build
whiteboards/markers/erasers, handout, coordinate axes in the ceiling, pumpkins, canned pineapple slices
description Student handout(PDF)
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 \((s,\phi,z+dz)\): \[d\vec{r}=\hspace{35em}\]
Path 3 from \((s,\phi,z)\) to \((s,\phi+d\phi,z)\): \[d\vec{r}=\hspace{35em}\]
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.
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 \(d\vec{r}\).
