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.
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 \((s,\phi+d\phi,z)\): \[d\vec{r}=\hspace{35em}\]
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.
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 \(d\vec{r}\).
In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.
This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..
This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector.
Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates.
The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.
Students work in groups to measure the steepest slope and direction on a plastic surface, and to
compare their result with the gradient vector, obtained by measuring its
components (the slopes in the coordinate directions).
Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral
\(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative.
They do a similar activity for the vector field \(\vec{G}\) which is not conservative.
Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.
Vector Calculus I 2022
You are climbing a hill along the steepest path, whose slope at your current location is \(1\over5\). There is another path branching off at an angle of \(30^\circ\) (\(\pi\over6\)). How steep is it?
Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
