Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
assignment Homework
Imagine you're standing on a landscape with a local topography described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\) is a constant, \(x\) and \(y\) are east and north coordinates, respectively, with units of kilometers. You're standing at the spot \((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). At the spot you're standing, what is the slope of the ground in the direction of the cottage? Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Does your result makes sense from the graph?
assignment Homework
Shown below is a contour plot of a scalar field, \(\mu(x,y)\). Assume that \(x\)
and \(y\) are measured in meters and that \(\mu\) is measured in kilograms.
Four points are indicated on the plot.
A contour map for a different function is shown above. On a printout of this contour map, sketch a field vector map of the gradient of this function (sketch vectors for at least 10 different points). The direction and magnitude of your vectors should be qualitatively accurate, but do not calculate the gradient for this function.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
assignment Homework
The electrostatic potential due to a point charge at the origin is given by: \begin{equation} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation}
assignment Homework
Find the gradient of each of the following functions:
assignment Homework
Consider the finite line with a uniform charge density from class.
group Small Group Activity
30 min.
assignment Homework
Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}
There is a hill in this classroom. The top of the hill is in the middle of the room at the ceiling. This topo map below describes the hill in the room. You are standing at some point on the topo map. Use your right arm to point in the direction of the gradient.
For this activity, the class is asked to stand from their seats. The students are told that they are all standing on an elliptical hill, represeted by the topo map, and one location of the classroom is selected as the top of a hill, typically in the center of the room. (If you are in a tiered lecture hall, then make use of the actual hill in the room, istead!)
The students are asked to close their eyse and point in the direction of the gradient.
Ask students to generalize the concepts in this activity to functions of three dimensions. Emphasize the understanding that the gradient is always perpendicular to the level curves (for two dimensions) or level surfaces (for three dimensions).
Reiterate the main points in "Student Conversations"