Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.
Each group is given a different two-dimensional vector \(\vec{k}\) and is asked to calculate the value of \(\vec{k} \cdot \vec {r}\) for each point on the grid and to draw the set of points with constant value of \(\vec{k} \cdot \vec{r}\) using rainbow colors to indicate increasing value.
On your whiteboard, there should be a 5x5 square grid of dots. The instructor will draw a specific vector \(\vec{k}\) on your grid.
For your \(\vec{k}\), connect dots with the same value of \(\vec{k} \cdot \vec{r}\).
Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.
Each group is given a different two-dimensional vector \(\vec{k}\) and is asked to calculate the value of \(\vec{k} \cdot \vec{r}\) for each point on the grid and to draw the set of points with constant value of \(\vec{k} \cdot \vec{r}\) using rainbow colors to indicate increasing value.
The group part of this activity should be quite quick, 5-10 minutes.
This is a compare and contrast activity. Ask each group to present. They should show their white board, show their vector \(\vec{k}\), and their curves of constant \(k\).
Points that should arise:
Questions to ask students:
Why are the lines of constant \(\vec{k} \cdot \vec{r}\) perpendicular to \(\vec{k}\)?
Usually someone in the class can come up with the explanation that all the position vectors \(\vec{r}\) that have the same projection onto \(\vec{k}\) have the same value of \(\vec{k} \cdot \vec{r}\).
This projection is easiest to interpret when I think about the dot product as \(\vec{k} \cdot \vec{r} = |\vec{k}|\color{blue}{|\vec{r}| \cos\theta}\), where \(|\vec{r}| \cos\theta\) is the projection of \(\vec{r}\) onto \(\vec{k}\).
This is a good time to remind the students that they have had to use two different representations of the dot product to completely understand this problem.
What if the grid of points was made a 3D cube of points? What would constant \(\vec{k} \cdot \vec{r}\) look like?
Constant \(\vec{k} \cdot \vec{r}\) are planes.
What does \(\cos(\vec{k} \cdot \vec{r})\) look like?
Planes that vary in value from -1 to 1 sinusoidally.
What does \(E_0\cos(\vec{k} \cdot \vec{r})\) look like?
Planes that vary in value from -\(E_0\) to \(E_0\) sinusoidally.
What does \(E_0\cos(\vec{k} \cdot \vec{r}-\omega t)\) look like?
Planes that move in the direction of \(\vec{k}\) at a rate of \(\omega/k\).
keyboard Computational Activity
120 min.
assignment Homework
Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).
keyboard Computational Activity
120 min.
finite difference hamiltonian quantum mechanics particle in a box eigenfunctions
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then usenumpy
to solve for eigenvalues and eigenstates, which they visualize.
assignment Homework
assignment Homework
group Small Group Activity
30 min.
assignment Homework
group Small Group Activity
30 min.
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.
assignment Homework
On the following diagrams, mark both \(\theta\) and \(\sin\theta\) for \(\theta_1=\frac{5\pi}{6}\) and \(\theta_2=\frac{7\pi}{6}\). Write one to three sentences about how these two representations are related to each other. (For example, see: this PHET)
assignment Homework
Consider the diagram of \(T\) vs \(V\) for several different constant values of \(p\).
Translate this diagram to a \(p\) vs \(V\) w/ constant \(T\) graph, including the point \(A\). Complete your graph by hand and make a fairly accurate sketch by printing out the attached grid or in some other way making nice square axes with appropriate tick marks.
Are the lines that you drew straight or curved? What feature of the \(TV\) graph would have to change to change this result?
Sketch the line of constant temperature that passes through the point \(A\).