Tangental 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%.
I think I got 19% which put me well in the top half of the class...
Introduction to Differential Equations (2008)
51–60 of 93 posts
Re: Introduction to Differential Equations (2008)
#52DiffEq 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.
You can do quite a lot simplifying to get equations that are solvable. A lot of engineering techniques do this, using simplifications that capture the essential behavior of a system (in fact I'd argue that knowing when such approximations are appropriate is one of the core skills of an engineer).
That said, even if you need a numerical solution it will still often require a lot of simplifications in order to be tractable. Multiphase fluid flow, for instance, relies on tons of physics simplifications and empirical correlations in order to make numerical techniques viable.
Re: Introduction to Differential Equations (2008)
#53Tangental 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
Re: Introduction to Differential Equations (2008)
#54DiffEq 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…
"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks…
I love #10, especially the last part. I would argue that even primary/secondary education is better served in this fashion.
Relevant to a recent Joe Rogan podcast with Neil Degrasse Tyson on how many teachers in your life inspired you. Not as much about the transfer of information, especially today, it's about the excitement and inspiration around the topics you are presenting.
Re: Introduction to Differential Equations (2008)
#55Oh wow, this site was an absolute godsend in college. I was stuck in a DiffEq class with an abysmal professor and barely hanging on. I found this a few weeks in, stopped bothering with useless lectures, and went from low-60s to high-80s over the rest of the semester.
Re: Introduction to Differential Equations (2008)
#56DiffEq 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…
I'd like to write a more algebraic understanding of differential equation, e, derivative and integrals. Something like Maxwell equations (pardon the pompous aspect) but for all things differentials. There's a webpage about differentiation that starts that (and is the inspiration behind my quest).
Re: Introduction to Differential Equations (2008)
#57Earlier quoted context omitted.
That's kind of where I was going with this. Instead of burning all of that time on symbolic differentiation, dig-down into numerical methods ASAP so students can get a feel for all of the related "gotchas"--of which there are many... edit: IMHO, many of those "gotchas" are much more interesting than the fundamentals of calculus.
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 that same time to apply the fundamentals to numerical methods, and you get to go in far more interesting directions--like symplectic integrals, or chaos theory.
Re: Introduction to Differential Equations (2008)
#58Earlier 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.
If you ask any STEM professor or TA that teaches undergrads, the reason so many students "wipe out" is because of a lack of preparation in fundamentals-- not just "calc 101", but even more basic than that, algebraic manipulation.
The material in a physics 101 course is just the barest minimum and it goes beautifully hand-in-hand with calculus 101.
Re: Introduction to Differential Equations (2008)
#59I took Cal 1 and 2 directly from Dr. Dawkins back in 2010-2011. He was the best teacher I had in college hands down. Totally changed the way I thought about math and I've been in love with it ever since.
Re: Introduction to Differential Equations (2008)
#60DiffEq 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…
Yeah, it's relatively dry material which is hard to grasp without the context of why we need it and how it can be applied in the real world. While the intro is good, it still has the same problem - just shows (in a good way) "some math".
We had a great professor and this was one of the most enjoyable classes I've ever taken. One particular assignment was a group paper where we were supposed to essentially explain and use the SIR model. We extended the model to an SIRZ model and effectively argued that zombie apocalypses in fiction are essentially impossible unless they include some supernatural elements. Under a wide range of assumptions (e.g., zombies rot/zombies don't rot) the infection always stopped before it spread significantly. (We got an A.)