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Introduction to Differential Equations (2008)

tutorial.math.lamar.edu

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Re: Introduction to Differential Equations (2008)

#41

Post by amelius »

hackandtrip wrote: post #34
amelius wrote: post #32

Earlier quoted context omitted.

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.

One of my professor used to say that you can do derivatives of every functions, it's mechanical, but you will only be able to find exact solution of an infinitesimal fraction of diffeq, and thinking about that, the same apply with integrals.

Yes the same applies to integrals. But when you go from 1d to 2d or 3d problems, the analytical approach quickly becomes too difficult.

Re: Introduction to Differential Equations (2008)

#42

Post by Cobord »

amelius wrote: post #32
acjohnson55 wrote: post #4

DiffEq 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.

Or you learn how to make the toy problem that has the same behavior as what you are trying to model. If you don't understand how to simplify problems, you end up just chucking the computation at it and calling it a day. If you have a solution to the basic problem you can give yourself a warm start.

Re: Introduction to Differential Equations (2008)

#43

Post by Cobord »

jstewartmobile wrote: post #37
crispyambulance wrote: post #36

Earlier quoted context omitted.

While numerical methods are absolutely critical in practice, analytic methods like you learn in what people call calculus and diff-eq are _absolutely_ essential to understanding the physical world. You can't actually _understand_ numerical methods without a fairly deep grounding in analytical methods. The real problem is here is a lack of context. Engineering and most science curriculums take a "short-cut" through ma…

> analytic methods ... are _absolutely_ essential to understanding the physical world How so? My experience has been that the "physical world" is where the symbolic approach completely breaks down. > Engineering and most science curriculums take a "short-cut" through mathematical education. Only people taking more math than scientists and engineers would be mathematicians. A year-and-a-half course to cover the limit,…

That is only if your allowing yourself to take the numerical methods as a black box. If you want to be sure that your method is okay, you prove error bounds as you take limits of 0 mesh size. Otherwise it is just building tables of black magic for what schema to use when.

Re: Introduction to Differential Equations (2008)

#44

Post by jstewartmobile »

Cobord wrote: post #43
jstewartmobile wrote: post #37

Earlier quoted context omitted.

> analytic methods ... are _absolutely_ essential to understanding the physical world How so? My experience has been that the "physical world" is where the symbolic approach completely breaks down. > Engineering and most science curriculums take a "short-cut" through mathematical education. Only people taking more math than scientists and engineers would be mathematicians. A year-and-a-half course to cover the limit,…

That is only if your allowing yourself to take the numerical methods as a black box. If you want to be sure that your method is okay, you prove error bounds as you take limits of 0 mesh size. Otherwise it is just building tables of black magic for what schema to use when.

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.

Re: Introduction to Differential Equations (2008)

#45

Post by sannee »

sideproject wrote: post #16

Does 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.

Re: Introduction to Differential Equations (2008)

#46

Post by crispyambulance »

jstewartmobile wrote: post #37
crispyambulance wrote: post #36

Earlier quoted context omitted.

While numerical methods are absolutely critical in practice, analytic methods like you learn in what people call calculus and diff-eq are _absolutely_ essential to understanding the physical world. You can't actually _understand_ numerical methods without a fairly deep grounding in analytical methods. The real problem is here is a lack of context. Engineering and most science curriculums take a "short-cut" through ma…

> analytic methods ... are _absolutely_ essential to understanding the physical world How so? My experience has been that the "physical world" is where the symbolic approach completely breaks down. > Engineering and most science curriculums take a "short-cut" through mathematical education. Only people taking more math than scientists and engineers would be mathematicians. A year-and-a-half course to cover the limit,…

    > "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, which give not only "an answer" but also tell you important features of the answer. These analytic solutions have "handles" you can use to ask "what-if" questions -- eg zero's in the denominator to indicate poles, behavior of the system as you take certain limits, geometric aspects such as symmetry, patterns in recurrence relations, etc, etc, etc..

Re: Introduction to Differential Equations (2008)

#47

Post by Cobord »

jstewartmobile wrote: post #44
Cobord wrote: post #43

Earlier quoted context omitted.

That is only if your allowing yourself to take the numerical methods as a black box. If you want to be sure that your method is okay, you prove error bounds as you take limits of 0 mesh size. Otherwise it is just building tables of black magic for what schema to use when.

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 table of what to use when without the generalizing principle behind it.

Re: Introduction to Differential Equations (2008)

#48

Post by jstewartmobile »

crispyambulance wrote: post #46
jstewartmobile wrote: post #37

Earlier quoted context omitted.

> analytic methods ... are _absolutely_ essential to understanding the physical world How so? My experience has been that the "physical world" is where the symbolic approach completely breaks down. > Engineering and most science curriculums take a "short-cut" through mathematical education. Only people taking more math than scientists and engineers would be mathematicians. A year-and-a-half course to cover the limit,…

> "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.

Re: Introduction to Differential Equations (2008)

#49

Post by _jal »

Cobord wrote: post #40
acjohnson55 wrote: post #4

DiffEq 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…

The algorithms you would learn for graphs etc are also just math and equally abstract. What makes the difference in how real you treat the two concepts that are both abstract but generally applicable?

Not the OP, but I've had good and bad math teachers. The bad ones tend to teach rote steps, "do this, do this, do this, done," without any attempt to explain why things work the way they do, without drawing parallels to already-learned things, without trying to teach any _why_.

Then you hear students asking, "when am I ever going to use this?"

My good teachers, on the other hand, always tied what we were doing into a larger scheme. If there were similarities or other relations between concepts, they'd be pointed out. If someone wasn't 'getting it', they had other ways of looking at it at hand, would sometimes give alternate methods of doing the same thing, etc.

In short, one tells you to memorize in a vacuum for no good reason. The other helps you learn.

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