Activity: Acting Out the Gradient

AIMS Maxwell AIMS 21 Static Fields Winter 2021
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".
What students learn
  • Geometric understanding of the directionality of the gradient vector.
  • Clarify misconceptions about the phrase “the gradient always points up hill”
  • Media
    • activity_media/hill.png
    • group Navigating a Hill

      group Small Group Activity

      30 min.

      Navigating a Hill
      AIMS Maxwell AIMS 21
    • assignment Directional Derivative

      assignment Homework

      Directional Derivative
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      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 Contours

      assignment Homework

      Contours
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      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 \((x,y)=(3,-2)\).
      4. 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 Number of Paths

      group Small Group Activity

      30 min.

      Number of Paths

      E&M Conservative Fields Surfaces

      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.
    • group Work By An Electric Field (Contour Map)

      group Small Group Activity

      30 min.

      Work By An Electric Field (Contour Map)

      E&M Path integrals

      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.
    • assignment The Gradient for a Point Charge

      assignment Homework

      The Gradient for a Point Charge
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      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.

    • assignment Gradient Practice

      assignment Homework

      Gradient Practice
      AIMS Maxwell AIMS 21

      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}

    • assignment Electric Field of a Finite Line

      assignment Homework

      Electric Field of a Finite Line

      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 Charged Sphere

      group Small Group Activity

      30 min.

      Charged Sphere

      E&M Introductory Physics Electric Potential Electric Field

      Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.
    • assignment Line Sources Using the Gradient

      assignment Homework

      Line Sources Using the Gradient
      AIMS Maxwell AIMS 21 Static Fields Winter 2021
      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}

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.

Instructor's Guide

Introduction

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.

Student Conversations

  1. Many students will incorrectly point towards the top of the hill, rather than perpendicular to the level curves. The gradient is not always the direction of the top of the hill. despite the gradient only lying in the x-y plane.
  2. Many students will incorrectly point upward. For a function of two variables, the gradient does not have a third, vertical component. The gradient lives in the topo map, not in 3-d space.

Wrap-up

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"


Author Information
Corinne Manogue, Tevian Dray
Learning Outcomes