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.
1. << Total Current, Square Cross-Section | Integration Sequence |
This activity is identical to Scalar Surface and Volume Elements except uses a more sophisticated vector approach to find directed surface, and volume elements.
This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins.
In a previous activity, Vector Differential--Curvilinear, students are asked to find the vector line element, \(d\boldsymbol{\vec{r}}\), along each side of an “infinitesimal box” in cylindrical and spherical coordinates. Using the \(d\boldsymbol{\vec{r}}\), they are now asked to construct the area (\(d\boldsymbol{\vec{A}}\)) and volume (\(dV\)) elements in each coordinate system. This prepares students to integrate vector- and scalar-valued functions in curvilinear coordinates.
Begin with a brief lecture which “derives” the formula \[d\boldsymbol{\vec{A}}=d\boldsymbol{\vec{r}}_1\times d\boldsymbol{\vec{r}}_2\] by drawing a differential area element on an arbitrary surface and appealing to the geometric definition of the cross product as a directed area. Label the sides of the surface element with vectors \(d\boldsymbol{\vec{r}}_1\) and \(d\boldsymbol{\vec{r}}_2\) with both vectors' tails at the same point.
Similarly, derive the formula \[d \tau=(d\boldsymbol{\vec{r}}_1\times d\boldsymbol{\vec{r}}_2)\cdot d\boldsymbol{\vec{r}}_3\] from a picture of an arbitrary differential volume element and the geometric definition of the scalar triple product as the volume of a parallelopiped. Label the sides of the volume element with vectors \(d\boldsymbol{\vec{r}}_1\), \(d\boldsymbol{\vec{r}}_2\), and \(d\boldsymbol{\vec{r}}_3\) with all the vectors' tails at the same point. Make sure to choose a right-handed orientation.
Next, ask the students to use these formulas to find the surface and volume elements for a plane, for a finite cylinder (including the top and bottom), and for a sphere.
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.