Divergence Theorem - Electrical Engineering Class Snap Video

I've become a big fan of what I call snap videos while teaching remotely. Here's a simple example I used recently to explain a concept that is relatively simple to describe in a traditional face-to-face class but maybe not so simple for students to understand while learning online.

Every undergraduate electrical engineering student needs to take an electromagnetics course. This course is a little different than most of the other required courses that use wires, resistors, capacitors, transistors etc, all physically (and in most cases two-dimensionally) connected together. 

In much of this course there are no wires and everything in three-dimensions - I sometimes refer to it as the "magic" course. The math is advanced calculus based but I've found if a student has made it far enough to be taking this course they've got the math down and can handle it. Conceptually is where they often stumble - trying to get a picture in their head of what is going on in three dimensions. 

The Divergence Theorem is a good example. In electromagnetics it is used to identify by location like sources and sinks. It is also used to explain the rate of change of a function with respect to position. Important stuff for things like cell and wifi signals along with a bunch of other "magic-based" technologies.

The math includes a couple methods called volume integration and surface integration. The volume integration is pretty easy - a student can bang through the math and get an answer without really having a good picture in their head of what is going on. Surface integration is a speed bump, wrong way turn, etc for many. I know it was for me when I was first learning this stuff. It really cannot be done without an accurate mental picture of what is going on. The classic way to introduce this topic uses a cube drawn in three dimensions on the board. Here's one of my (not so good) drawings in three dimensions (x, y and z axis) as an example. 

The cube (yeah, that's a cube!) has all six sides labeled and to solve the problem students need to do surface integration math on each of the six sides individually and then combine the six answers for a final answer. Which side is which is where the confusion lies - which on is side 1?? Looking at my drawing above - I can't figure it out.... My diagram is pretty much useless!!

When teaching in the classroom I hold up a box and describe and label the different sides with the students. Can't do that online so..... how about a video. Here's a quick one I put together a couple days ago, describing and hopefully explaining the confusing parts. 

I'm using an Apple iPad with Apple Pencil along with the GoodNotes app. I find it useful to "think out loud" when I do these. It is also the way I teach - thinking through a problem step by step with the students. I do not do any editing so this 6 minute video took maybe 10 minutes total to record and upload to YouTube.