Gradient Sequence

This sequence starts with an introduction to partial derivatives and continues through gradient. While some of the activities/problems are pure math, a number of other activities/problems are situated in the context of electrostatics. This sequence is intended to be used intermittently across multiple days or even weeks of a course or even multiple courses.
1. Activity: Partial Derivatives from a Contour Map
In this sequence of small whiteboard questions, students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?
2. Homework Problem: Contours
Follow-up homework in which students calculate (estimate) partial derivatives from a contour map and use these numbers to plot examples of the gradient vector.

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.

  1. Determine \(\frac{\partial\mu}{\partial x}\) and \(\frac{\partial\mu}{\partial y}\) at each of the four points.
  2. On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the gradient of \(\mu(x,y)\) using your answers to part a above. How did you choose a scale for your vectors? Describe how the direction of the gradient vector is related to the contours on the plot and what property of the contour map is related to the magnitude of the gradient vector.
  3. Evaluate the gradient of \(h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3\) at the point \((x,y)=(3,-2)\).

3. Activity: The Hill
This is one of our favorite activities. We've used it successfully in multi-variable and vector calculus classes, electrostatics classes, and workshops on active engagement. Follow it with the kinesthetic activity Acting Out the Gradient
In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
4. Activity: Acting Out the Gradient
A great kinesthetic activity to follow up on The Hill.
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".
5. Activity: The Hillside
Optional. Requires plastic surfaces, see The Surface Project Overview.
Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
6. Activity: Visualising the Gradient
Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
7. Activity: Directional Derivatives
Optional. Requires plastic surfaces, see The Surface Project Overview.
This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.
8. Homework Problem: Directional Derivative
Follow-up homework on Directional Derivative.

You are on a hike. The altitude nearby is 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 \((1~\mathrm{km}, 2~\mathrm{km})\). You drop your water bottle and the water spills out.

  1. Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
  2. In which direction in space does the water flow?
  3. At the spot you're standing, what is the slope of the ground in the direction of the cottage?
  4. Does your result to part (c) make sense from the graph?

9. Homework Problem: The Path
Follow-up homework on Directional Derivative.
You are climbing a hill along the steepest path, whose slope at your current location is \(1\over5\). There is another path branching off at an angle of \(30^\circ\) (\(\pi\over6\)). How steep is it?
10. Homework Problem: Gradient Practice
Students are required to calculate the gradient of pure math functions expressed in rectangular, cylindrical, and spherical coordinates.

Find the gradient of each of the following functions:

  1. \begin{equation} f(x,y,z)=e^{(x+y)}+x^2 y^3 \ln \frac{x}{z} \end{equation}
  2. \begin{equation} \sigma(\theta,\phi)=\cos\theta \sin^2\phi \end{equation}
  3. \begin{equation} \rho(s,\phi,z)=(s+3z)^2\cos\phi \end{equation}

11. Homework Problem: Gradient Point Charge
A straightforward electrostatics homework problem that demonstrating the relationship between potential and electric field for a point charge. Requires students to calculate a gradient in rectangular coordinates.

Consider the fields at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\).

  1. Write down an expression for the electrostatic potential \(V(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
  2. Write down an expression for the electric field \(\vec{E}(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
  3. Working in rectangular coordinates, compute the gradient of \(V\).
  4. Write several sentences comparing your answers to the last two questions.

12. Homework Problem: The Gradient for a Point Charge
An electrostatics homework problem that requires students to calculate a gradient in rectangular, cylindrical, and spherical coordinates. Beware, this problem is challenging since student will need to right down position vectors in these coordinate systems.

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}

  1. Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
  2. Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
  3. Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.

13. Homework Problem: Line Sources Using the Gradient
A straightforward electrostatics homework problem that requires students to calculate a gradient in cylindrical coordinates.
  1. 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}