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## Gradient: Instructor's Guide

### Main Ideas

- Computationally modeling the gradient
- Visualizing the geometric relationship between the gradient and a scalar field
- Clarification of how gradients look in two and three dimensions.
- Increased familiarity with Maple and/or Mathematica

### Students' Task

*Estimated Time: 10 - 15 minutes*

See files on activity page.

### Prerequisite Knowledge

A rudimentary understanding how a gradient acts on a scalar field.

### Props/Equipment

### Activity: Introduction

This activity is a great follow up to acting out the gradient.

The students should first look at all the graphs inside the maple/mathematica files. Then have the students plot some functions of their own.

### Activity: Student Conversations

### Activity: Wrap-up

### Extensions

This activity is part of a sequence of activities which address the Geometry of Vector Fields. The following activities are included in this sequence.

- Preceding activity:
- Curvilinear Basis Vectors: This kinesthetic activity students are asked to point in $\hat{r}$, $\hat{\theta}$, $\hat{\phi}$, and $\hat{z}$ directions in reference to an origin within the classroom which begins class discussion regarding about the directions of curvilinear basis vectors at various points in space.

- Follow-up activities:
- Drawing Electric Field Vectors: In direct analogy to Drawing Equipotential Surfaces, this small group activity has students sketch the electric field due to a quadrupole in the plane of the charges.
- Visualizing Electric Flux: This computer visualization activity uses Mathematica to explore the effects of placing a point charge inside, outside, and on a cubical Gaussian surface which allows students to visualize the electric flux of a point charge through a Gaussian surface in different locations with respect to the point charge.
- Visualizing Divergence: This computer visualization activity has students predict the sign of divergence at various points in many vector fields generated by a Mathematica notebook.
- Visualizing Curl: Similar to Visualizing Divergence, this activity uses a Mathematica notebook of various vector fields to assist students in the geometric interpretation of the curl of a vector field by predicting the sign of the curl at various points in vector fields.

This activity is part of a sequence of activities which address the Geometry of the Gradient. The following activities are included within this sequence.

- Preceding activities:
- Acting Out the Gradient: This kinesthetic activity introduces students to the geometric concept of the gradient through an imaginary elliptic hill in the classroom where students use their arms to represent the gradient at their local point within the classroom.
- Navigating a Hill: 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. This activity emphasizes the gradient as a local quantity and requires students to perform calculations on a given scalar field.