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.

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