A Great Moment With Spherical Coordinates

In my multivariable calculus class yesterday, I was introducing spherical and cylindrical coordinates. These students didn’t take precalculus, so they didn’t have a ton of practice with polar coordinates. So I thought it was going to be a tough class to teach. To compound that, the smartboard wasn’t working, so I couldn’t show them some great demonstrations (here and here).

We had to do things by hand.

And I wanted to spend the entire period working on deriving the conversions:

Given a point in rectangular coordinates, how do you find the cylindrical coordinates?
Given a point in cylindrical coordinates, how do you find the rectangular coordinates?
Given a point in rectangular coordinates, how do you find the spherical coordinates?
Given a point in spherical coordinates, how do you find the rectangular coordinates?

And we went through it all. It wasn’t easy, but they “saw” it. They came up with the conversions. I just guided them.

I had two favorite moments in this class:

(1) At one point, when dealing with spherical coordinates, one student discovered an equation for the angle made with the positive z-axis. The equation wasn’t the one in the book. I suspected the two were the same, but decided I wasn’t going to pursue it. A student said we couldn’t leave it hanging, we HAD to figure it out. So we did, together. That 7 minute aside was awesome.

(2) The school clocked stopped at one point, and of course, it was noticed by a student. (Do my students wait with hungry eyes for the course to end?) Apparently the clocks in the school do that occasionally. And somehow, in a fit of impulsivity, I got onto an aside about standardizing time, Einstein’s work in the Patent Office in Switzerland, and his original 1905 paper on special relativity. (See Peter Galison’s Einstein’s Clocks, Poincare’s Maps if you want a lengthy version of what I told them.)

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