Activity: Acting Out the Gradient

Static Fields 2022 (6 years)
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

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"

  • group The Hill

    group Small Group Activity

    30 min.

    The Hill
    Vector Calculus II 2022 (4 years)

    Gradient

    Gradient Sequence

    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.
  • group DELETE Navigating a Hill

    group Small Group Activity

    30 min.

    DELETE Navigating a Hill
    Static Fields 2022 (5 years) 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.
  • group The Hillside

    group Small Group Activity

    30 min.

    The Hillside
    Vector Calculus I 2022

    Gradient Sequence

    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).
  • assignment Directional Derivative

    assignment Homework

    Directional Derivative

    Gradient Sequence

    Static Fields 2022 (6 years)

    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?

  • group Directional Derivatives

    group Small Group Activity

    30 min.

    Directional Derivatives
    Vector Calculus I 2022

    Directional derivatives

    Gradient Sequence

    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.
  • assignment Gradient Point Charge

    assignment Homework

    Gradient Point Charge

    Gradient Sequence

    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.

  • assignment The Path

    assignment Homework

    The Path

    Gradient Sequence

    Vector Calculus I 2022 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?
  • assignment Contours

    assignment Homework

    Contours

    Gradient Sequence

    Static Fields 2022 (6 years)

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

  • 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.

Author Information
Corinne Manogue, Tevian Dray
Learning Outcomes