Activity: 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.

• This activity is used in the following sequences
What students learn
• How to find area, and volume elements in curvilinear coordinates using geometric methods.

Instructor's Guide

Notes:

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

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

Introduction

In a previous activity, Vector Differential--Curvilinear, students are asked to find the line element, $d\ell$, along each side of an “infinitesimal box” in cylindrical and spherical coordinates. Using the $d\ell$, they are now asked to construct the area ($dA$) and volume ($dV$) elements in each coordinate system. This prepares students to integrate scalar-valued functions in curvilinear coordinates.

Find the formulas for the differential surface $dA$ and volume $d\tau$ elements (little chopped pieces of the surface and/or volume) for a plane, for a finite cylinder (including the top and bottom), and for a hemisphere. Make sure to draw an appropriate figure.

Learning Outcomes