## Activity: The Hillside

Vector Calculus I 2022
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).
• This activity is used in the following sequences
What students learn
• The gradient is a geometric representation of the steepest slope and direction.

Warm-up: Imagine you are standing on the side of a tall hill. List three things you would want to know about your position.

On your Mark: Place your surface on the grid. Label the $x$ and $y$ directions on the grid and surface. Measure the slope in the direction of greatest increase of the surface at the blue dot. Include units.

Slope in steepest direction: $\underline{\hspace{2in}}$

Get Set: The surface's height $h$ is a function of $x$ and $y$, giving $h = h(x,y)$. At the blue dot, measure both $\frac{\partial h}{\partial x}$ and $\frac{\partial h}{\partial y}$. Then form the vector $\frac{\partial h}{\partial x}\,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y}\,\boldsymbol{\hat{y}}$. Include units.

$\frac{\partial h}{\partial x} = \underline{\hspace{.75in}} \hspace{.5in} \frac{\partial h}{\partial y} = \underline{\hspace{.75in}} \hspace{.5in} \frac{\partial h}{\partial x} \,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y} \,\boldsymbol{\hat{y}} = \underline{\hspace{.75in}} \,\boldsymbol{\hat{x}} + \underline{\hspace{.75in}} \,\boldsymbol{\hat{y}}$

Go: At the blue dot, which way does your vector $\frac{\partial h}{\partial x}\,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y}\,\boldsymbol{\hat{y}}$ point on the surface?

1. What is your vector's magnitude?
2. How does your vector relate to the level curve through the blue dot?

Challenge: Rotate the surface $30^\circ$ on the grid and redraw the $x$ and $y$ directions on your surface. Which of your answers to On your Mark, Get Set, and Go remain the same?

Copyright 2014 by The Raising Calculus Group

## Instructor's Guide

### Main Ideas

• The gradient is a geometric representation of the steepest slope and direction.

Students work in groups to measure the steepest slope and direction, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

### Prerequisite Knowledge

• Familiarity with measuring partial derivatives using surfaces
• Basic knowledge of vectors

### Props/Equipment

• Surface models and contour maps
• A handout for each student

### Introduction

No introduction is needed, provided the prerequisite content (partial derivatives and vectors) has been covered. In particular, it is not necessary to have discussed the gradient yet.

### Student Conversations

Encourage students to measure their slopes as accurately as possible. (The correct slope should be approximately 1.16 in each case!)

Some groups will forget to keep track of signs, especially if the slopes in the $x$- and $y$-directions are different.

Ask each group to draw their gradient vector on their contour map. This allows the instructor to quickly determine whether their answers are correct, as they gradient should be perpendicular to the level curves.

Some students may need to be prodded to draw these vectors in the right place on the contour maps, as some students may use the contour maps merely as a convenient writing surface, without correctly aligning their surface with the map. Ask such students where the blue dot is on the map.

Be very careful as to not lead students to believe that all vectors' magnitudes depend on their slopes. This misunderstanding was common in a later activity.

Students may not understand why the gradient vector is perpendicular to the level curve passing through the point in question.

### Wrap-up

Ask students to present their final drawing, showing the gradient vector on the map. (A document camera can be useful for this purpose.) Most students will quickly realize that their vector was indeed perpendicular to the level curves, and some should also realize that their computed magnitude (based on the partial derivatives in the coordinate directions) agrees reasonably well with their measured slope in the steepest direction.

### Extensions

A natural followup activity is the [[courses:activities:vcact:vchill|Hill]] activity. A part of this latter activity is to emphasize that the gradient vector in these examples is in fact horizontal; it has no $\boldsymbol{\hat{z}}$-component. (This fact can also be brought out in the wrap-up, without using the Hill activity.)

Similarly, the Hill activity emphasizes that the gradient vector does not point toward the top of the hill, which could also be brought out during the wrap-up.

Both of these features activity can be brought out by asking students to stand up, imagining themselves standing on the contour diagram for the Hill activity, close their eyes, and stick their right hand out in the direction of the gradient vector at their location. (The instructor should designate one corner of the room to represent the center of the diagram.) Many students will incorrectly point “up”, while some may incorrectly point to the top.

• group Directional Derivatives

group Small Group Activity

30 min.

##### Directional Derivatives
Vector Calculus I 2022

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 DELETE Navigating a Hill

group Small Group Activity

30 min.

##### 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 The Hill

group Small Group Activity

30 min.

##### The Hill
Vector Calculus II 23 (4 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.
• assignment The Path

assignment Homework

##### The Path

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?
• 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 Directional Derivative

assignment Homework

##### Directional Derivative

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?

computer Mathematica Activity

30 min.

Static Fields 2023 (6 years)

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.

assignment Homework

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.

• accessibility_new Acting Out the Gradient

accessibility_new Kinesthetic

10 min.

Static Fields 2023 (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".
• assignment_ind Possible Worldlines

assignment_ind Small White Board Question

10 min.

##### Possible Worldlines
Theoretical Mechanics (4 years)

Student consider several curves on a spacetime diagram and have to judge which curves could be worldlines for an object.

Author Information
Surfaces team
Learning Outcomes