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On a related note, how do you take notes for math subjects with their multiline integrals and sumation and subscripts, division etc? I'm very much attached to recording every thing in simple text editors. Is there a "Notepad" or "TextEdit" for mathematical notation?
While the linked course seems a lot more thorough, I took the Udacity "Differential Equations in Action" [1] course, which I found very well done. For the homework you write Python programs to compute things like gravitational slingshots, modeling epidemics, wildfires, and the n-body problem. [1] https://www.udacity.com/course/differential-equations-in-act...
I took the first part of that course, enough to write an n-body game: https://github.com/madengr/nbody While it was a good course, I wish it covered the analytic portion, rather than dive straight into numerical solutions.
'Initial' refers to time-like variables, and 'boundary' refers to space-like variables.
There is no notion of time in mathematics. There is only a notion of space, due to geometry. I had the longest time coming to grips with the question, "mathematically what is the difference between initial value and boundary value?" only to realize the distinction is meaningless in mathematics. It's a relic of the past when differential equations were studied under physics, where time and space are a huge part of the conceptual foundation.
Sometimes I wonder how much progress we would make in education if we didn't confuse the heck of our students in the name of convention and historical baggage.
Lately I have been looking for a practical (not too much theory) book on how to model problems with differential equations, with lots of examples. I imagine I'd plug them into Wolfram Alpha to get a solution. Does anyone have recommendations?
You can also look into "system dynamics" software, which allows for the modeling of complex systems with very little user-facing mathematics. Dana Meadows wrote a book called "Thinking in Systems," which is a great intro to this subject.¹ I personally have used Stella by isee systems, which will output an equation from your model if you are so inclined, but I think there is cheaper/free software out there that will d…
The usage of terms 'initial value' and 'boundary value' is a massive failure of mathematics education. 'Initial' refers to time-like variables, and 'boundary' refers to space-like variables. There is no notion of time in mathematics. There is only a notion of space, due to geometry. I had the longest time coming to grips with the question, "mathematically what is the difference between initial value and boundary valu…
Look at Arnold's book on Diff, it is harder then others, but has absolutely different outlook.
https://www.amazon.com/Ordinary-Differential-Equations-MIT-P...
Earlier quoted context omitted.
I took the first part of that course, enough to write an n-body game: https://github.com/madengr/nbody While it was a good course, I wish it covered the analytic portion, rather than dive straight into numerical solutions.
That is pretty neat. Improvements on the analytic side (if you are interested in the opinion of a random person on the internet) would be to scale your coordinate system (e.g. scale the distance and time to make average velocity and radius equal 1) and look into using a symplectic integrator. Better for these types of systems.
Look at Arnold's book on Diff, it is harder then others, but has absolutely different outlook.
I can only find the first edition on Amazon, but there's apparently a third edition? http://www.matem.unam.mx/ernesto/EDO/Ar-EDO.pdf https://www.amazon.com/Ordinary-Differential-Equations-MIT-P...
On a related note, how do you take notes for math subjects with their multiline integrals and sumation and subscripts, division etc? I'm very much attached to recording every thing in simple text editors. Is there a "Notepad" or "TextEdit" for mathematical notation?
Earlier quoted context omitted.
> I really don't think that such topics can be self-studied Can you explain this? What makes something extremely difficult to be studied without a teacher? (I took calc in high school and never took diff eq, so my knowledge in this specific domain is basically zero)
Diff. eq. is a difficult topic. Not only you need to have a good understanding of a lot of other mathematic topics (derivatives, integrals etc) to properly understand it since it builds upon those but you also need somebody to guide you on how to think when solving such problems (i.e you'll need to solve diff. eq. together with your teacher).