Solving the heat equation: Differential equations, chapter 3 [video] | Hacker News Reader

Solving the heat equation: Differential equations, chapter 3 [video]

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For an odd application of PDE: you can define and solve boundary value problems over spatial domains in games to find a solution over the playing field that measures how near or far you are from desireable or undesireable locations. Then objects in the game (e.g. actors controlled by AI) can use simple gradient ascent / gradient descent using the local gradient of the solution to determine how to move, in order to e.g. avoid threats or chase the player

https://sgd.cs.colorado.edu/wiki/Collaborative_Diffusion

Poisson's Equation is also used for mesh reconstruction: https://en.m.wikipedia.org/wiki/Poisson%27s_equation#Surface...

I recently read a book on the life of Faraday and Maxwell[1]. Very interestingly, Maxwell modelled the famous equations of Electromagnetism by imagining them to be similar to how heat flows. To me that is a great take-away: Trying to model a phenomenon based on an existing phenomenon (within reason, of course) and changing the world with it!

[1]https://www.goodreads.com/book/show/18240286-faraday-maxwell...

Edit: some formatting

I read this book too and can also recommend it.

Here's a small toy to play with the heat equation: https://math.berkeley.edu/~kmill/toys/heat/heat.html (it probably doesn't work on mobile---sorry!)

It shows the Fourier transform of the curve you draw, and how each summand decays through time.

There's also a wave equation toy: https://math.berkeley.edu/~kmill/toys/wave/wave.html

3B1B is one of those content creators that consistently puts me in awe at how much stuff is out there today that is both very informative and very well produced.

I love the subjects they treat and the teaching style; however, a video is useless to me, i can't follow an argument in video format for some reason. If there were textual transcripts of the same words with static illustrations (or a few, non-automatic animations) it would be much more useful.

It sounds like what you're asking for is a paper textbook.

It can be a website also. Animations are cool, as long as they are not overwhelming.

I'm the same. I mostly watch to get some sort of satisfaction thinking I'm "not really" wasting my time, since whatever I'm watching might be useful for me... which it never proves to be.

Some are visual thinkers, I'm more intuitive.

I think the visualizations are amazing for bringing concepts together, but you also absolutely have to do the work if you want to really understand and continue to work with the math concepts they talk about.

I haven't really used 3b1b to advance my math understanding (I used it for math entertainment), but if I did, I would watch the video, work stuff out on paper, and then watch the video again

I'm not sure what it is about this video I don't like. It just didn't seem very much like a 3B1B video for some reason. Maybe it's because there were fewer "A HA" moments than in other videos. Maybe its the source material. Just feels...different.

Not sure why you did not have any "Aha" moment, bu the visual explanation used for the principle of superposition and approximation of functions by Fourier series is very informative. I also liked the explanation about the proportional relation between temperature and space in the heat equation. I think in most higher educational systems, they only approach this type of problems with pure theoretical solutions/explanations.

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