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## Visualizing Divergence: Instructor's Guide

### Main Ideas

- divergence as the flux per unit volume through an infinitesimal box
- sign and relative magnitude of flux at various points in space
- checking predictions with Maple/Mathematica

### Students' Task

*Estimated Time: 20 min*

Students look at several vector fields and calculate their divergence to get a sense of what a non-zero divergence might look like.

This worksheet is designed to be an instructor-led activity. You would need to add appropriate instructions and questions to use this as an independent student activity. The activity can be used quite effectively with the instructor projecting the worksheet at the front of the room if students do not have access to a computer for each small group.

### Prerequisite Knowledge

- Some familiarity with Maple
- Introduction to vector fields
- Algebraic understanding of divergence

### Props/Equipment

### Activity: Introduction

We precede this activity with a derivation of the rectangular expression for divergence from the definition that divergence is the flux per unit volume through an appropriately chosen closed surface. Our derivation follows the one in “Div, grad, curl and all that”, Schey, 2nd edition, Norton, 1973, p. 36. One can also use clicker questions or SWBQs about divergence to help get them started (see reflections).

The worksheet shows a number of different vector fields. Most vector fields are shown as a cross-section of the field and it is assumed the the vector field is independent of the third (unshown) dimension. Students are asked to use the definition of divergence as the flux per unit volume through an infinitesimal box to predict the sign and relative magnitude of the flux at various points in the vector field. The worksheet then calculates the divergence, so students can check their predictions.

### Activity: Student Conversations

**Choosing the right shape chunks**Students should be encouraged to see that it is easier to choose a surface that respects the symmetries of the vector field, i.e. pineapple chunks for cylindrical fields, pumpkin chunks for spherical fields etc.

**Various points**Make sure to look at several different points in space for each vector field, not just the origin. Use this to emphasize that divergence is itself a field which, when combined with Gauss' Law, tells you what the charge distribution is at each point in space.

**Positive or negative divergence**Students should should see that, for the vector field that radiates out from the origin, different length scalings lead to different signs for the divergence, depending on whether they are adding larger vectors along the larger arced surface or smaller vectors along the larger arced surface. Near the end of this activity, they can be asked to discover which scaling leads to zero divergence everywhere (except at the origin). This vector field represents the electric field around a charged wire. Nature picks out this special case.

### Activity: Wrap-up

No particular wrap-up is needed.

### Extensions

This activity pairs nicely with the Visualizing Curl activity.

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 activities:
- 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.
- Visualizing Gradient: This activity uses Mathematica to show the gradient of several scalar fields.
- 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.

- Follow-up activities:
- 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.