Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
1. << The Hillside | Gradient Sequence | Directional Derivatives >>
Use the Sage code at Using Technology to Visualize the Gradient to plot scalar functions of two variables, predict what a plot of the gradient looks like, and then check it.
This activity is a great follow up to Acting Out the Gradient. The students should know that the gradient is perpendicular to level curves (surfaces) and that its magnitude is the maximum slope. They use this information to predict the gradient vector field from looking at level curves for scalar fields. Then check their predictions with the Mathematica worksheet. The worksheet is open-ended and students can plot some scalar fields of their own.
computer Mathematica Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
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.
accessibility_new Kinesthetic
10 min.
group Small Group Activity
30 min.
coulomb's law electric field charge ring symmetry integral power series superposition
Students work in small groups to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
format_list_numbered Sequence