Students use known algebraic expressions for vector line elements \(d\vec{r}\) to determine all simple vector area \(d\vec{A}\) 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.
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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, \(d\vec{r}\), along each side of an “infinitesimal box” in cylindrical and spherical coordinates. Using the \(d\vec{r}\), they are now asked to construct the area (\(d\vec{A}\)) 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 \[d\vec{A}=d\vec{r}_1\times d\vec{r}_2\] 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 \(d\vec{r}_1\) and \(d\vec{r}_2\) with both vectors' tails at the same point.
Similarly, derive the formula \[d \tau=(d\vec{r}_1\times d\vec{r}_2)\cdot d\vec{r}_3\] 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 \(d\vec{r}_1\), \(d\vec{r}_2\), and \(d\vec{r}_3\) 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.
group Small Group Activity
30 min.
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.
assignment Homework
Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.
group Small Group Activity
5 min.
assignment Homework
The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}
This problem explores the consequences of the divergence theorem for this shell.
groups Whole Class Activity
10 min.
There are two versions of this activity:
As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.
As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distribued to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.
group Small Group Activity
30 min.
charge charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.group Small Group Activity
30 min.
vector calculus coordinate systems curvilinear coordinates
In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).
Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.
keyboard Computational Activity
120 min.
quantum mechanics operator matrix element particle in a box eigenfunction
Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.assignment Homework
List variables in their proper positions in the middle columns of the charts below.
Solve for the magnetic susceptibility, which is defined as: \[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T \]
Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:
\[\left(\frac{\partial M}{\partial B}\right)_S \]
Evaluate your chain rule. Sense-making: Why does this come out to zero?
group Small Group Activity
30 min.