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## Scalar Distance, Area, and Volume Elements: Instructor's Guide

### Main Ideas

- Line, area, and volume elements in curvilinear coordinates

### Students' Task

*Estimated Time: *

- Determine all possible $ds$, $dA$, and $d\tau$ in cylindrical and spherical coordinates

Students are asked to find the differential expressions for the following:

- The area element of a plane in rectangular coordinates for $z={\rm const}$.
- The area element of a plane in polar coordinates for $z={\rm const}$.
- The area element of the side of a cylinder ($r={\rm const}$) in cylindrical coordinates.
- The area element of the top of a cylinder ($z=z_{top}={\rm const}$) in cylindrical coordinates.
- The area element of the bottom of a cylinder ($z=z_{bottom}={\rm const}$) in cylindrical coordinates.
- The area element of the surface of a sphere in spherical coordinates ($r={\rm const}$).
- The volume element of a block in rectangular coordinates.
- The volume element of a cylinder (a “pineapple chunk”) in cylindrical coordinates.
- The volume element of a sphere (a “pumpkin piece”) in spherical coordinates.

This activity works well as a Compare and Contrast activity, with different groups solving different cases and then reporting their results to the class as a whole.

### Prerequisite Knowledge

- Some familiarity with curvilinear coordinates
- Integration from previous calculus courses

### Props/Equipment

- Tabletop Whiteboard with markers

### Activity: Introduction

Begin this activity with a review of cylindrical and spherical coordinates which will likely be a review for most students. This activity is identical to Surface and Volume Elements in Cylindrical and Spherical Coordinates except uses a scalar approach to find line, surface, and volume elements. Therefore, this activity does not require knowledge of the $d\vec{r}$ vector, cross products, and dot products which can make this activity more accessible to students earlier in a course on electricity and magnetism. In the activity, students are asked to find the line element, $ds$, along each side of an “infinitesimal box” in cylindrical and spherical coordinates. Using $ds$, they are then asked to construct the area ($dA$) and volume ($dV$) elements in each coordinate system. This prepares students to integrate in curvilinear coordinates.

This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins.

### Activity: Student Conversations

### Activity: Wrap-up

Students should have obtained the following common surface and volume elements:

$dA= dx\,dy$ for a plane with $z={\rm const}$ in rectangular coordinates.

$dA =r\,dr\,d\phi$ for a plane with $z={\rm const}$ in polar coordinates.

$dA=r\,dr\,d\phi$ for the top of a cylinder with $z={\rm const}$.

$dA=r\,dr\,d\phi$ for the bottom of a cylinder with $z={\rm const}$.

$dA=r\,d\phi\, dz$ for the side of a cylinder with $r={\rm const}$.

$dA=r^2\sin\theta\,d\theta\,d\phi$ for the surface of a sphere with $r={\rm const}$.

$d\tau = dx\,dy\,dz$ a small block rectangular coordinates.

$d\tau = r\,dr\,d\theta$ for a “pineapple chuck” in cylindrical coordinates.

$d\tau = r^2\,sin{\theta}\,dr\,d\theta\,d\phi$ for a “pumpkin piece” in spherical coordinates.

### Extensions

This is the initial activity within a sequence of activities addressing Scalar Integration in Curvilinear Coordinates. The following activities are included within this sequence:

- Preceding activities:
- Internal Energy of Derivative Machine: This small group activity uses a modified Partial Derivative Machine to measure the internal energy of a nonlinear, one dimensional system while emphasizing integration as an experimentally measurable quantity.
- Curvilinear Coordinates: This lecture introduces students to curvilinear coordinates and highlights the notation difference of $\theta$ and $\phi$ in physics and mathematics.

- Follow-up activities:
- Pineapples and Pumpkins: This activity can be done in small groups or as an instructor led whole class activity where a pineapple (for cylindrical) and/or pumpkin (for spherical) can be cut to demonstrate the geometry of an infinitesimal volume element used in integration.
- Acting Out Charge Densities: This kinesthetic activity provides students with an embodied understanding of charge density and total charge by using their bodies to represent charges and act out linear, surface, and volume charge densities which prompts a whole class discussion on the meaning of constant charge density, the geometric differences between linear, surface, and volume charge densities, and what is “linear” about linear charge density.
- Total Charge: In this small group activity, students calculate the total charge within spherically or cylindrically symmetric volumes by using multivariable integration in various coordinate systems in order to find the total charge contained within the volume due to a specific charge density.