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>But from that, you can ask what is the one dimensional analogue of this conservation of mass principle and you get .. the fundamental theorem of calculus! Thi

by floxy 6y ago

>But from that, you can ask what is the one dimensional analogue of this conservation of mass principle and you get .. the fundamental theorem of calculus! Things like this sound awesome! Is there a book you can recommend? Or are you writing this book? Because I want to pre-order it.

rsj_hn 6y ago
The book was written by Arnol'd. I recommend reading his opus magnum, written mostly as he was commuting on the Moscow Subway. It's called "The Mathematical Methods of Classical Mechanics". https://www.amazon.com/Mathematical-Classical-Mechanics-Graduate-Mathematics/dp/0387968903 https://www.amazon.com/Mathematical-Classical-Mechanics-Grad... but be careful, it is a bit terse. You have to spend a lot of time with it and some paper and pencil, working things out. The reason why Gauss's principle is just a generalization of the fundamental theorem of calculus is that this general result is that Integral over the boundary = Integral over the interior of the divergence, or more poetically Int_(dA)A = Int_A dA Assume you have some fluid flowing down the number line, where f(t) is the amount of fluid flowing through t. And this number line has some fluid sources and sinks in (things that add or subtract fluid). For an incompressible fluid, you will only get more fluid at f(t+h) then you have at f(t) if there some fluid producing source between t and t+h that adds a bit of fluid, df, to the total. So the total amount of fluid flowing past b will be the fluid that enters the interval at a, f(a), together with the sum over all the divergences (sources) between a and b. Thus f(b) = Int(df) + f(a). The reason the one dimensional analogue of divergence is just the derivative should be clear enough, the divergence is the rate of change in all directions (gradient) but in one dimension, the gradient is just the derivative. In fact you can prove the multi-dimensional version from the one dimensional version via slicing and applying the one dimensional argument, taking into account the linear properties of the gradient (e.g. rate of change along some vector given by the sum of directions a + b is the sum of the partial derivatives along a and b). I unfortunately am not writing any books, I am cranking out code for work and hot takes on hackernews for fun. I wish I had time to write a book, but I have often fantasized about writing math books for kids, especially parents homeschooling kids, but it could be anyone. I would also recommend the following (Russian) books by Kolmogorov and Aleksandrov: https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163 https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
- floxy 6y ago
  Thanks for the recommendation. I'd probably need the undergraduate version unfortunately. But this internet stranger wants to encourage you write the book for kids someday.

by floxy 6y ago

rsj_hn 6y ago

The book was written by Arnol'd. I recommend reading his opus magnum, written mostly as he was commuting on the Moscow Subway. It's called "The Mathematical Methods of Classical Mechanics". https://www.amazon.com/Mathematical-Classical-Mechanics-Graduate-Mathematics/dp/0387968903 https://www.amazon.com/Mathematical-Classical-Mechanics-Grad... but be careful, it is a bit terse. You have to spend a lot of time with it and some paper and pencil, working things out. The reason why Gauss's principle is just a generalization of the fundamental theorem of calculus is that this general result is that Integral over the boundary = Integral over the interior of the divergence, or more poetically Int_(dA)A = Int_A dA Assume you have some fluid flowing down the number line, where f(t) is the amount of fluid flowing through t. And this number line has some fluid sources and sinks in (things that add or subtract fluid). For an incompressible fluid, you will only get more fluid at f(t+h) then you have at f(t) if there some fluid producing source between t and t+h that adds a bit of fluid, df, to the total. So the total amount of fluid flowing past b will be the fluid that enters the interval at a, f(a), together with the sum over all the divergences (sources) between a and b. Thus f(b) = Int(df) + f(a). The reason the one dimensional analogue of divergence is just the derivative should be clear enough, the divergence is the rate of change in all directions (gradient) but in one dimension, the gradient is just the derivative. In fact you can prove the multi-dimensional version from the one dimensional version via slicing and applying the one dimensional argument, taking into account the linear properties of the gradient (e.g. rate of change along some vector given by the sum of directions a + b is the sum of the partial derivatives along a and b). I unfortunately am not writing any books, I am cranking out code for work and hot takes on hackernews for fun. I wish I had time to write a book, but I have often fantasized about writing math books for kids, especially parents homeschooling kids, but it could be anyone. I would also recommend the following (Russian) books by Kolmogorov and Aleksandrov: https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163 https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

floxy 6y ago
Thanks for the recommendation. I'd probably need the undergraduate version unfortunately. But this internet stranger wants to encourage you write the book for kids someday.