## Activity: Vector Differential--Curvilinear

Vector Calculus II 2022 (7 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.

• This activity is used in the following sequences
• Media

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.

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.

## 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

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 $d\vec{r}$.

• assignment_ind Vector Differential--Rectangular

assignment_ind Small White Board Question

10 min.

##### Vector Differential--Rectangular
Static Fields 2022 (7 years)

Integration Sequence

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..

• group Vector Integrals (Contour Map)

group Small Group Activity

30 min.

##### Vector Integrals (Contour Map)

• group Number of Paths

group Small Group Activity

30 min.

##### Number of Paths

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.
• group Work By An Electric Field (Contour Map)

group Small Group Activity

30 min.

##### Work By An Electric Field (Contour Map)

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.
• group Covariation in Thermal Systems

group 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.
• group Quantifying Change (Remote)

group Small Group Activity

30 min.

##### Quantifying Change (Remote)

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.
• accessibility_new Time Dilation Light Clock Skit

accessibility_new Kinesthetic

5 min.

##### Time Dilation Light Clock Skit

Students act out the classic light clock scenario for deriving time dilation.
• assignment Paramagnet (multiple solutions)

assignment Homework

##### Paramagnet (multiple solutions)
Energy and Entropy 2021 (2 years) We have the following equations of state for the total magnetization $M$, and the entropy $S$ of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\} \end{align}
1. List variables in their proper positions in the middle columns of the charts below.

2. Solve for the magnetic susceptibility, which is defined as: $\chi_B=\left(\frac{\partial M}{\partial B}\right)_T$

3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

$\left(\frac{\partial M}{\partial B}\right)_S$

4. Evaluate your chain rule. Sense-making: Why does this come out to zero?

• group Paramagnet (multiple solutions)

group Small Group Activity

30 min.

##### Paramagnet (multiple solutions)
• Students evaluate two given partial derivatives from a system of equations.
• Students learn/review generalized Leibniz notation.
• Students may find it helpful to use a chain rule diagram.
• assignment Free Expansion

assignment Homework

##### Free Expansion
Energy and Entropy 2021 (2 years)

The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.*

The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between $p$, $V$ and $T$. You may take the number of molecules $N$ to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.
1. What is the change in entropy of the gas? How do you know this?

2. What is the change in temperature of the gas?

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