Wow, it's been a while. Does anyone actually use this stuff after college???
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Introduction to Differential Equations (2008)
61–70 of 93 posts
Re: Introduction to Differential Equations (2008)
#62Does anyone know websites or resources which explains how Diff Eq is used in Computer Science? I know it's used in a variety of areas in CS, but I really like to see or read well-explained tutorials or articles. e.g. what's finding area got to do with the topics in CS? what does 'area' correspond to?
I took DiffEQs (the only 5-credit course offered, which I thought rather odd), of course, on my way to my CP degree at FIT in central Florida, and the one thing I remember was that they turn out to be very useful in biological simulations.
As I remember, population ebb-and-flow based upon available resources is something it was especially good at estimating and tracking over time, so I would imagine that any of the Sims-type games they would be quite useful.
It turns out they are damn good at estimating, over time, many things, so a little bit of research and the right game and it's not hard to see how the two could work well together.
Re: Introduction to Differential Equations (2008)
#63Earlier quoted context omitted.
> "physical world" is where the symbolic approach completely breaks down. It does not "break down" it just becomes intractable in certain cases. One needs to be able to solve problems that have all but the most essential details stripped out in order to develop a sense of how physical law actually works. Many times that is even "good enough" to get to a solution. The best way to do that is through analytic methods, w…
I would posit that the reason so many people wipe-out in undergrad physics is that the coursework insists on pounding the square peg of law into the round hole of analytic methods. Something most people in the STEM fields refuse to acknowledge is that throwing away information complicates things just as often as it simplifies them. I say that if the ball doesn't bounce forever, the equation should reflect that.
It entirely depends on your goals. If you want to be a productive engineer, then most of that stuff is not going to be useful in your job.
OTOH, if you study physics at an advanced level, it's rather shocking how effectively all that analysis models the world, despite throwing away a lot of information. Try studying solid state physics. It's crazy the number of assumptions they make, and yet the theory still produces very accurate results.
There's a reason Eugene Wigner penned an essay with the title "The Unreasonable Effectiveness of Mathematics".
Re: Introduction to Differential Equations (2008)
#64Earlier quoted context omitted.
Consider symplectic integrators. You would never come up with them or realize the problem of energy drift if you hadn't first paid attention to the fundamentals of the geometry and calculus underlying the problem. This is just my favorite example, but it illustrates how understanding the fundamentals also explains the gotchas. Just getting a feel for them through experience is again just black magic by building up a…
Never said to do away with the fundamentals. Will say that most symbolic differentiation and integration (which is a big chunk of the coursework) is not fundamental as much as it is fruitless busywork. Even so, I spent a year of my time--and God knows how much of other people's money--grinding out the mathematical equivalent of crossword puzzles so I could get my job certificate--just like every other engineer. Use t…
>Never said to do away with the fundamentals. Will say that most symbolic differentiation and integration (which is a big chunk of the coursework) is not fundamental as much as it is fruitless busywork.
It's no more busywork than being able to multiply two single digit numbers in your head. Whether it's useful to your job really depends on the job. I had a job once in the engineering industry. When we were in meetings discussing projects, if you could not do those types of analyses (e.g. asymptotic behavior of certain Calc II type integrals) in your head, you would not know what's going on. Sure, everyone could explicitly show all the steps for your benefit, but you'd be slowing everyone down.
Re: Introduction to Differential Equations (2008)
#65I think the schools spend far too much time on symbolic differentiation and integration. This limits the exercises to the kinds of toy problem that yield to those methods. Kids get sidetracked on solving anti-differentiation puzzles, while the fundamentals are relegated to those (largely useless) puzzles. After 20 years of engineering--in almost every case--numerical methods have been the only way forward. In hindsig…
Theoretical fluid dynamicist here. I solve a lot of not nice differential equations exactly or approximately.
Whether numerical methods are viewed as the primary way forward is a bit of a self-fulfilling prophecy. If you don't think analytical solutions end up being useful, you probably won't put in the work needed to generate them in the first place, so you never see the value.
Even if you go all in with numerical methods, you need to test your code. This requires an exact solution and knowledge of the convergence rate of the numerical scheme. The exact solution can be for a special case that is easy to solve. You might need multiple exact solutions to cover all the physics. You can also use techniques like the method of manufactured solutions, but if you don't like analytical methods you'd probably hate that.
You need to check if the empirical convergence rate matches the theoretical one. In practice this is rarely done, but it's essential towards eliminating bugs. So you can't entirely avoid exact solutions if you want to do purely numerics right. This was not covered in my first differential equations class, unfortunately, but I think it's an essential topic.
Exact solutions are often impossible, but less so than most people believe. I've produced exact solutions many times to equations people thought required numerics. The exact solutions are very valuable by themselves, as they can be used much faster than numerical solutions in most cases and allow you to see the structure of the solution. I think you should always try hard to make an analytical exact or approximate solution. It might be rare that you can do it, but the value is large and if we stopped teaching these methods it would become much more rare.
As for you mentioning in another post the problem of "pounding the square peg of law into the round hole of analytic methods", you should learn about approximate analytical solutions, which give you a lot more flexibility. You still ultimately have the same problem, though.
Re: Introduction to Differential Equations (2008)
#66Earlier quoted context omitted.
I would posit that the reason so many people wipe-out in undergrad physics is that the coursework insists on pounding the square peg of law into the round hole of analytic methods. Something most people in the STEM fields refuse to acknowledge is that throwing away information complicates things just as often as it simplifies them. I say that if the ball doesn't bounce forever, the equation should reflect that.
It entirely depends on your goals. If you want to be a productive engineer, then most of that stuff is not going to be useful in your job. OTOH, if you study physics at an advanced level, it's rather shocking how effectively all that analysis models the world, despite throwing away a lot of information. Try studying solid state physics. It's crazy the number of assumptions they make, and yet the theory still produces…
Word up for Mr. Wigner !!
Re: Introduction to Differential Equations (2008)
#67Tangental anecdote: Every time I see Diff EQ mentioned the first thing that pops into my head is the number 11. That's the score of my first, last, and only Diff EQ test. 11%.
had you studied at all? most undergrad differential equations classes are fairly mechanical in nature, you just learn to identify the type of problem, then you follow the steps exactly as they are written in the textbook, super little variation or freedom
This is often more true than many believe. But in my experience few students get so far that they recognize the patterns and can respond in the way you describe. That might be the crux.
I remember my undergraduate mechanical vibrations class. Every exam was basically a test of how well you could do the Laplace transform on some linear ODEs. I memorized the most common transforms, so this became fairly straightforward and fast for me, but it was obvious the other students were struggling.
If I had a problem that wasn't solveable with the Laplace transform, say a linear ODE with variable coefficients, I'd likely take longer to do the exam, but those never appeared in the class.
Re: Introduction to Differential Equations (2008)
#68Does anyone know websites or resources which explains how Diff Eq is used in Computer Science? I know it's used in a variety of areas in CS, but I really like to see or read well-explained tutorials or articles. e.g. what's finding area got to do with the topics in CS? what does 'area' correspond to?
Not really Computer Science per se, but practically every engineering field uses numerical solvers. No one reallistically solves nontrivial differential equations by hand these days.
Not so much in engineering, but in science, approximate perturbation techniques are still a big deal.
Re: Introduction to Differential Equations (2008)
#69DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…
Also, what they usually don't tell you is that the recipe bag only works for toy problems. For real applications you most often need numerical approximations.
If you consider analyzing an RL/RC circuit as a "toy problem" then I guess you're right.
Re: Introduction to Differential Equations (2008)
#70Does anyone know websites or resources which explains how Diff Eq is used in Computer Science? I know it's used in a variety of areas in CS, but I really like to see or read well-explained tutorials or articles. e.g. what's finding area got to do with the topics in CS? what does 'area' correspond to?
Discrete difference equations show up everywhere, and are basically the same stuff. Knuth-Oren-Patashnik's Concrete Mathematics is great