Student handout: Vector Surface and Volume Elements

AIMS Maxwell 2021

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.

What students learn Students use \(d\boldsymbol{\vec{A} }= d\boldsymbol{\vec{r}}_1 \times d\boldsymbol{\vec{r}}_2\) and \(d\tau=(d\boldsymbol{\vec{r}}_1\times d\boldsymbol{\vec{r}}_2)\cdot d\boldsymbol{\vec{r}}_3\) to find differential surface and volume elements for cylinders and spheres.
Find the formulas for the differential surface and volume elements for a plane, for a finite cylinder (including the top and bottom), and for a sphere. Make sure to draw an appropriate figure.
Learning Outcomes