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Activities

Small Group Activity

30 min.

Vector Surface and Volume Elements

Students use known algebraic expressions for vector line elements \(d\boldsymbol{\vec{r}}\) to determine all simple vector area \(d\boldsymbol{\vec{A}}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

  • Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Small Group Activity

30 min.

Scalar Surface and Volume Elements

Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

  • Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Small Group Activity

30 min.

Curvilinear Volume Elements
Students construct the volume element in cylindrical and spherical coordinates.
  • Found in: Vector Calculus I course(s)

Small Group Activity

30 min.

A glass of water
Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.

You have a charge distribution on the \(x\)-axis composed of two point charges: one with charge \(+3q\) located at \(x=-d\) and the other with charge \(-q\) located at \(x=+d\).

  1. Sketch the charge distribution.
  2. Write an expression for the volume charge density \(\rho (\vec{r})\) everywhere in space.

Problem

5 min.

Volume Charge Density

Consider the volume charge density: \begin{equation*} \rho (x,y,z)=c\,\delta (x-3) \end{equation*}

  1. (2 pts) Describe in words how this charge is distributed in space.
  2. (2 pts) What are the dimensions of the constant \(c\)?

  • Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)

Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

  1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
  2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
  3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

  • Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)

Whole Class Activity

10 min.

Pineapples and Pumpkins

There are two versions of this activity:

As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area and volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distributed to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

  • Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Kinesthetic

10 min.

Acting Out Charge Densities
Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and we demonstrate the answers with coins under a doc cam.

Kinesthetic

10 min.

Acting Out Current Density
Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.

Small Group Activity

30 min.

Vector Differential--Curvilinear

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.

Small Group Activity

30 min.

Visualization of Divergence
Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the geometric definition of the divergence of a vector field at a point as flux per unit volume (here: area) through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
  • Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Flux Sequence sequence(s)

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

30 min.

Heat and Temperature of Water Vapor
In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.

Small Group Activity

30 min.

The Cylinder
This small group activity is designed to help students visual the process of chopping, adding, and multiplying in single integrals. Students work in small groups to determine the volume of a cylinder in as many ways as possible. The whole class wrap-up discussion emphasizes the equivalence of different ways of chopping the cylinder.
  • Found in: Vector Calculus I course(s)