Summer of Math Exposition

Big thanks for the video! Feedback: 1. I think that the visual component of the video is great, and that everything that could be shown visually was nicely animated. The transition from 3d column to the cone, then to the section of a circle is very clear. The motivation for the introduction of polar coordinates is therefore also abundantly clear. 2. In the beginning of the video you mentioned that polar coordinates make the whole range of different problems approachable, but this hasn't been followed up at all in the video. It would be nice to show where else this topic shows up, maybe even introduce the spherical coordinates and their applications in geography and astronomy. In short, the video is begging to be continued in some way. 3. Perhaps the main formulas on 5:12 deserve short explanations, especially given that they constitute the desired transformation from one coordinate system to another. But I agree that it's simple to understand these formulas, especially having all the terms illustrated visually.

Great stuff! It's a genuine pleasure to watch a SoME video that is (a) accessible to a very wide audience, (b) explained very thoughtfully and skillfully, and (c) food for thought to those who are already familiar with the mathematics involved. Lovely graphics, too. Most of the glory in these contests has tended to go to more advanced mathematics, but I really like this one. Thanks for making it.

I would have expected a general coordinates, not just polar coordinates when you say you want to generalise this problem, going into differential geometry.

Nice animations, good visualization of the coordinate mapping. Motivation is taken very seriously and done well. Focused video with the right length.

I loved how familiar ideas were used to build on the explanation, which fell naturally. I also enjoyed the further reading available. This was a novel way of understanding polar coordinates and together with the proof, this makes for a memorable and enjoyable maths piece.

What's an animation, I loved the animation, even the video feel really simple for me, i think your annimation style is just amazing.

A nice example of an application of polar coordinates. Very clearly and enjoyably explained. Nice graphics too. Well done.

'Molar subordinates' sounds like something from chemistry. Bonus points for conciseness, clarity and brevity Only minor quibble I have is, having explained this for grid->cone, you should have at least mentioned how you might expand the technique to cover arbitrary curved shapes - i.e general uv mapping

Animations were amazing. Development was well motivated and exposition was clear. The only thing I am left wondering as a teacher if there is a context (perhaps more modern) that would give rise to the same math, instead of carving a picture onto a column. Perhaps a problem in computer animation?

Very nice framing of the problem to introduce polar coordinates as a way that we can solve the problem. I think what I liked most about this was how we took a problem and extended it slowly to something that we could easily understand. I'd probably like a bit more discussion on how we derive the radius of the larger circle a bit as that is probably an interesting topic to also dive into.