Activity: DELETE Navigating a Hill

Static Fields 2023 (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 Small Group Activity schedule 30 min. build Tabletop Whiteboard with markers,Computers with Maple (optional),A handout for each student description Student handout (PDF)
What students learn
  • The gradient is perpendicular to the level curves.
  • The gradient is a local quantity, i.e. it only depends on the values of the function at infinitesimally nearby points.
  • Although students learn to chant that "the gradient points uphill," the gradient does not point to the top of the hill.
  • The gradient path is not the shortest path between two points.

Suppose you are standing on a hill. You have a topographic map, which uses rectangular coordinates \((x,y)\) measured in miles. Your global positioning system says your present location is at one of the following points (pick one): \[A:(1,4),\quad B:(4,-9),\quad C:(-4,9),\quad D:(1,-4),\quad E:(2,0),\quad F:(0,3)\] Your guidebook tells you that the height \(h\) of the hill in feet above sea level is given by \[h=a-bx^2-cy^2\] where \(a=5000~\mathrm{ft}\), \(b=30\,\mathrm{\frac{ft}{mi^2}}\), and \(c=10\,\mathrm{\frac{ft}{mi^2}}\).

  1. Starting at your present location, in what map direction (2-dimensional unit vector) do you need to go in order to climb the hill as steeply as possible? Draw this vector on your topographic map.
  2. How steep is the hill if you start at your present location and go in this compass direction? Draw a picture which shows the slope of the hill at your present location.

Instructor's Guide

Introduction

We preface this activity with a mini-lecture about the gradient. Students should be familiar with how to calculate a gradient:

\[\boldsymbol{\vec{\nabla}} f = \frac{\partial f}{\partial x}\>\boldsymbol{\hat{x}}+ \frac{\partial f}{\partial y}\>\boldsymbol{\hat{y}}+\frac{\partial f}{\partial z}\>\boldsymbol{\hat{z}}\]

and the geometric property that the gradient points in the direction of greatest increase in the function.

Student Conversations

  • Students very quickly figure out the location of the top of the hill and the height of the hill.
  • Where does the gradient live? Students will not realize that, because the height is a function of two variables in this problem, the gradient of the height function is a 2-D vector that lives in the topo map.
  • Compass direction versus slope: The gradient tells the students the direction of steepest ascent, and it also contains information about how quickly the height function is changing. \[df = \vec{\nabla}f\cdot d\vec{r}\] Therefore, the magnitude of the gradient is the slope. To find the 3-D vector direction of travel, students need to find a unit vector in the direction of gradient as well as the change in height. Most students will forget that they need a normalized vector in the x-y plane to give the 3-D vector pointing along the steepest direction at their point.

Wrap-up

The key understanding is that the gradient is always perpendicular to the level curves(for two dimensions) that they lie on. This is regardless of where the global maximum is of the function: if the level curves are not circular, then not everyone will be pointing towards the same point.

Have the students stand up and imagine the room is a hill, with the top of the hill in one of the corners of the room. Have the students close their eyes and use their rights arms to point in the direction of the gradient. Students should be alerted to the fact that their arms should be parallel to the floor. (See this activity.)
  • group The Hill

    group Small Group Activity

    30 min.

    The Hill
    Vector Calculus II 23 (7 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 The Hillside

    group Small Group Activity

    30 min.

    The Hillside
    Vector Calculus I 2022 (2 years)

    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).
  • accessibility_new Acting Out the Gradient

    accessibility_new Kinesthetic

    10 min.

    Acting Out the Gradient
    Static Fields 2023 (7 years)

    gradient vector fields electrostatics

    Gradient Sequence

    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 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 Directional Derivative

    assignment Homework

    Directional Derivative

    Gradient Sequence

    Static Fields 2023 (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?

  • assignment The Gradient for a Point Charge

    assignment Homework

    The Gradient for a Point Charge

    Gradient Sequence

    Static Fields 2023 (6 years)

    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

    Gradient Sequence

    Static Fields 2023 (4 years)

    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 Contours

    assignment Homework

    Contours

    Gradient Sequence

    Static Fields 2023 (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 \((x,y)=(3,-2)\).

Learning Outcomes