## Activity: Acting Out the Gradient

Static Fields 2022 (4 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

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 Navigating a Hill

group Small Group Activity

30 min.

##### Navigating a Hill
Static Fields 2022 (3 years)
• assignment Directional Derivative

assignment Homework

##### Directional Derivative
Static Fields 2022 (4 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 Contours

assignment Homework

##### Contours
Static Fields 2022 (4 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

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)

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
Static Fields 2022 (4 years)

The electrostatic potential due to a point charge at the origin is given by: $$V=\frac{1}{4\pi\epsilon_0} \frac{q}{r}$$

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 Homework

Static Fields 2022 (3 years)

Find the gradient of each of the following functions:

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

• 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

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
Static Fields 2022 (4 years)
1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: $$V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right)$$

Author Information
Corinne Manogue, Tevian Dray
Learning Outcomes