Static Fields 2022 (6 years)
Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
• This activity is used in the following sequences
What students learn
• Model the gradient using a computer algebra system.
• Visualize the geometric relationship between the gradient and the contours plot of a scalar field
• Clarify how gradients look in two and three dimensions.

Use the Sage code at Using Technology to Visualize the Gradient to plot scalar functions of two variables, predict what a plot of the gradient looks like, and then check it.

## Instructor's Guide

This activity is a great follow up to Acting Out the Gradient. The students should know that the gradient is perpendicular to level curves (surfaces) and that its magnitude is the maximum slope. They use this information to predict the gradient vector field from looking at level curves for scalar fields. Then check their predictions with the Mathematica worksheet. The worksheet is open-ended and students can plot some scalar fields of their own.

• computer Using Technology to Visualize Potentials

computer Mathematica Activity

30 min.

##### Using Technology to Visualize Potentials
Static Fields 2022 (6 years)

Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrates several different ways of plotting the potential.
• 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 The Hill

group Small Group Activity

30 min.

##### The Hill
Vector Calculus II 2022 (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.
• 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.
• assignment Contours

assignment Homework

##### Contours

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

• accessibility_new Acting Out the Gradient

accessibility_new Kinesthetic

10 min.

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".
• 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 The Hillside

group Small Group Activity

30 min.

##### 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).

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