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.
1. << Vector Differential--Rectangular | Integration Sequence | Pineapples and Pumpkins >>
Cylindrical Coordinates:Find the general form for \(d\vec{r}\) in cylindrical coordinates by determining \(d\vec{r}\) along the specific paths below.
- Path 1 from \((s,\phi,z)\) to \((s+ds,\phi,z)\): \[d\vec{r}=\hspace{35em}\]
- Path 2 from \((s,\phi,z)\) to \((s,\phi+d\phi,z)\): \[d\vec{r}=\hspace{35em}\]
- Path 3 from \((s,\phi,z)\) to \((s,\phi,z+dz)\): \[d\vec{r}=\hspace{35em}\]
If all three coordinates are allowed to change simultaneously, by an infinitesimal amount, we could write this \(d\vec{r}\) for any path as:
\[d\vec{r}=\hspace{35em}\]
This is the general line element in cylindrical coordinates.
Spherical Coordinates:Find the general form for \(d\vec{r}\) in spherical coordinates by determining \(d\vec{r}\) along the specific paths below.
- Path 1 from \((r,\theta,\phi)\) to \((r+dr,\theta,\phi)\): \[d\vec{r}=\hspace{35em}\]
- Path 2 from \((r,\theta,\phi)\) to \((r,\theta+d\theta,\phi)\): \[d\vec{r}=\hspace{35em}\]
- Path 3 from \((r,\theta,\phi)\) to \((r,\theta,\phi+d\phi)\): (Be careful, this is a tricky one!) \[d\vec{r}=\hspace{35em}\]
If all three coordinates are allowed to change simultaneously, by an infinitesimal amount, we could write this \(d\vec{r}\) for any path as:
\[d\vec{r}=\hspace{35em}\]
This is the general line element in spherical coordinates.
This activity allows students to derive formulas for \(d\vec{r}\) in cylindrical, and spherical coordinates, using purely geometric reasoning. These formulas form the basis of our unified view of all of vector calculus, so this activity is essential. For more information on this unified view, see our publications, especially: Using differentials to bridge the vector calculus gap
Using a picture as a guide, students write down an algebraic expression for the vector differential in different coordinate systems (cylindrical, spherical).
Begin by drawing a curve (like a particle trajectory, but avoid "time" in the language) and an origin on the board. Show the position vector \(\vec{r}\) that points from the origin to a point on the curve and the position vector \(\vec{r}+d\vec{r}\) to a nearby point. Show the vector \(d\vec{r}\) and explain that it is tangent to the curve.
It may help to do activity Vector Differential--Rectangular as an introduction.
For the case of cylindrical coordinates, students who are pattern-matching will write \(d\vec{r} = dr\, \hat{r} + d\phi\, \hat{\phi} + dz\, \hat{z}\). Point out that \(\phi\) is dimensionless and that path two is an arc with arclength \(r\, d\phi\).
Some students will remember the formula for arclength, but many will not. The following sequence of prompts can be helpful.
For the spherical case, students who are pattern matching will now write \(d\vec{r} = dr\, \hat{r} + d\phi\, \hat{\phi} + d\theta\, \hat{\theta}\). It helps to draw a picture in cross-section so that they can see that the circle whose arclength gives the coefficient of \(\hat{\theta}\) has radius \(r\sin\theta\). It can also help to carry around a basketball to write on to talk about the three dimensional geometry of this problem.
The only wrap-up needed is to make sure that all students have (and understand the geometry of!) the correct formulas for \(d\vec{r}\).
assignment_ind Small White Board Question
10 min.
vector differential rectangular coordinates math
In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.
This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
accessibility_new Kinesthetic
5 min.
Special Relativity Time Dilation Light Clock Kinesthetic Activity
Students act out the classic light clock scenario for deriving time dilation.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.
assignment Homework
The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.^{*}
The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between \(p\), \(V\) and \(T\). You may take the number of molecules \(N\) to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.What is the change in entropy of the gas? How do you know this?