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Activities

Small Group Activity

60 min.

The Park

This is the first activity relating the surfaces to the corresponding contour diagrams, thus emphasizing the use of multiple representations.

Students work in small groups to interpret level curves representing different concentrations of lead.

None
  • Found in: AIMS Maxwell, Static Fields, Problem-Solving course(s)

Small Group Activity

30 min.

Vector Differential--Curvilinear

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 \((s,\phi,z+dz)\): \[d\vec{r}=\hspace{35em}\]
  • 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.

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 \(d\vec{r}\).

This mini-lecture demonstrates the relationship between \(df\) on the tangent plane to its “components“ in coordinate directions, leading to the multivariable chain rule.

Small Group Activity

30 min.

Covariation in Thermal Systems
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.

Small Group Activity

60 min.

Multivariable Pictionary
Students draw the 3D graphs of equations using three variables. They make choices for drawing a stack of curves in parallel planes and a curve in a perpendicular plane (e.g. substituting in values for \(x\), \(y\), or \(z\). )

Small Group Activity

30 min.

Quantifying Change
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.

Small Group Activity

30 min.

Chain Rule Measurement
This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the \(r\)-direction and derivatives in \(x\)- and \(y\)-directions using the chain rule.
  • Found in: Vector Calculus I course(s)

Small Group Activity

5 min.

The Resistors
This small group activity is designed to provide practice with the multivariable chain rule. Students determine a particular rate of change using given information involving other rates of change. The discussion emphasizes the equivalence of a variety of approaches, including the use of differentials. Good “review” problem; can also be used as a homework problem.
  • Found in: Vector Calculus I, Surfaces/Bridge Workshop course(s)

Small Group Activity

5 min.

Finding a Chain Rule
Students use chain rule diagrams to construct a multivariable chain rule in terms of differentials.
  • Found in: Surfaces/Bridge Workshop course(s)