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I discovered a new use for small white boards this week. I was explaining to my (small, honors) vector calculus class about adapted bases, and we had reached spherical coordinates. As I now usually do, I had everyone stand up, close their eyes, and put their arms out along r-hat, theta-hat and phi-hat, with theta-hat being the hardest. But for the first time, I put the origin on the ceiling, rather than the floor, so that the students were in the Southern Hemisphere. This was challenging even for me, so I looked for a way to make it clear.

We had been working with small white boards, so I asked the class to use their boards to show me where the tangent plane to the sphere would be at their location. My class is already accustomed to my asking questions using these boards, so I guess I shouldn't have been surprised: They all started drawing pictures of spheres on their boards, while still standing up.

“No,” I said, “use the board to represent the tangent plane.”

“Oh,” they said.

I was then able to ask whether $\hat \theta$ (and $\hat \phi$) should be in this plane (yes), and where $\hat r$ should be (perpendicular to this plane). Despite the initial confusion, I believe the boards helped to solidify these concepts for the students, and will happily repeat the experience.

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The first curvilinear unit vectors are very confusing, but after a discussion of “unit vector as direction in which coordinate is changing” my students did well. But I also found theta-hat in spherical coordinates completely threw my class, even after they did well with phi hat and cylindrical coordinates. Perhaps this is because we were in the northern hemisphere, so theta-hat was down. I did not try putting the origin on the ceiling, as described above, but that would be good practice.

Also, this activity is fun.