format_list_numbered Sequence

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.

computer Mathematica Activity

30 min.

Visualising the Gradient
Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
  • Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Gradient Sequence sequence(s)

accessibility_new Kinesthetic

10 min.

Acting Out the Gradient
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".

group Small Group Activity

30 min.

DELETE Navigating a Hill
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.
  • Found in: Static Fields, AIMS Maxwell course(s)

group Small Group Activity

30 min.

Directional Derivatives
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.

group Small Group Activity

30 min.

The Hillside
Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
  • Found in: Vector Calculus I course(s) Found in: Gradient Sequence sequence(s)

group Small Group Activity

30 min.

The Hillside (Updated)
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).
  • Found in: Surfaces/Bridge Workshop, Problem-Solving course(s)

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}

  • Found in: Gradient Sequence sequence(s) Found in: AIMS Maxwell, Static Fields, Problem-Solving course(s)

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.

  • Found in: Gradient Sequence sequence(s)

group Small Group Activity

30 min.

The Hill
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.
  • Gradient
    Found in: Vector Calculus II, Vector Calculus I, Surfaces/Bridge Workshop course(s) Found in: Gradient Sequence sequence(s)

None

Contours

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 \((x,y)=(3,-2)\).

  • Found in: Gradient Sequence sequence(s) Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)

group Small Group Activity

30 min.

Number of Paths
Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral \(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative. They do a similar activity for the vector field \(\vec{G}\) which is not conservative.

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.

  • Found in: Gradient Sequence sequence(s) Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)

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?

  • Found in: Gradient Sequence sequence(s) Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)

Consider the finite line with a uniform charge density from class.

  1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
  2. Perform the integral to find the \(z\)-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the \(s\)-component as well!)

group Small Group Activity

30 min.

Work By An Electric Field (Contour Map)
Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

group Small Group Activity

5 min.

Maxima and Minima
This small group activity introduces students to constrained optimization problems. Students work in small groups to optimize a simple function on a given region. The whole class wrap-up discussion emphasizes the importance of the boundary.
  • Found in: Vector Calculus I course(s)

group Small Group Activity

60 min.

The Valley
Students compute vector line integrals and explore their properties.
  • Found in: Vector Calculus II, Surfaces/Bridge Workshop course(s)

group Small Group Activity

30 min.

Murder Mystery Method
  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}

  • Found in: Gradient Sequence sequence(s) Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)