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Introduction to Differential Equations (2008)

tutorial.math.lamar.edu

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Re: Introduction to Differential Equations (2008)

#81

Post by jshowa1 »

amelius wrote: post #79
jshowa1 wrote: post #69

Earlier quoted context omitted.

If you consider analyzing an RL/RC circuit as a "toy problem" then I guess you're right.

Linear circuits are mostly analyzed using the Laplace transform, i.e. in the s-domain, where the differential equations are abstracted away. In the time-domain, simulators are still used most often. But yes a really simple circuit like RC/RL is usually done on the back on the envelope, but then you're talking really simple. The problem with the analytical approach to differential equations is that it doesn't scale we…

Laplace transformations are differential equations so I fail to see your point. They're just in a different domain. However I do see your point with numerical methods since most complex problems are simulated anyways through simulating software. So in essence, the application becomes pointless because its at such a higher level of abstraction that you don't even have to think about it. You just punch in some numbers and hit analyze and the computer does it all for you.

Re: Introduction to Differential Equations (2008)

#82

Post by amelius »

jshowa1 wrote: post #81
amelius wrote: post #79

Earlier quoted context omitted.

Linear circuits are mostly analyzed using the Laplace transform, i.e. in the s-domain, where the differential equations are abstracted away. In the time-domain, simulators are still used most often. But yes a really simple circuit like RC/RL is usually done on the back on the envelope, but then you're talking really simple. The problem with the analytical approach to differential equations is that it doesn't scale we…

Laplace transformations are differential equations so I fail to see your point. They're just in a different domain. However I do see your point with numerical methods since most complex problems are simulated anyways through simulating software. So in essence, the application becomes pointless because its at such a higher level of abstraction that you don't even have to think about it. You just punch in some numbers…

> Laplace transformations are differential equations so I fail to see your point.

What I mean is that typically an electrical engineer will convert L and C elements to complex impedances (which depend on the frequency through s), and will then compute as though the elements are ordinary resistances. The expression "d/dt" isn't used in the entire analysis.

See: https://en.wikipedia.org/wiki/Phasor

Quoting:

> the phasor transform thus allows the analysis (calculation) of the AC steady state of RLC circuits by solving simple algebraic equations (albeit with complex coefficients) in the phasor domain instead of solving differential equations (with real coefficients) in the time domain

Re: Introduction to Differential Equations (2008)

#83

Post by freetime2 »

jshowa1 wrote: post #72
freetime2 wrote: post #19

Earlier quoted context omitted.

"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…

"Such a teacher will be successful if at the end of the course every one of his or her students feels they have taken “a good course,” even though they may not quite be able to pin down anything specific they have learned in the course." Basically the equivalent of, if the course makes you feel good, then the course is good. Not something that I'd advocate. By the way, a differential equation is simply an equation wi…

Actually, I have degree in math from a reputable university. So while you are certainly entitled to think that isn't going "far enough in math", I do think that your opinion is in the minority here. If anything, I would say the fact that I have completely forgotten everything from that class is evidence I may have actually gone too far in math... or further than I ended up needing.

And I think it's evidence that the differential equations are not taught in a way that is beneficial for comp sci students (and other types of students too, but I can't speak to that). I took other classes that have not been applicable to my career after graduation - things like finite state machines, computability, and complexity theory. But I still remember a lot from those classes - due to their focus on fundamental ideas and proving things.

Re: Introduction to Differential Equations (2008)

#84

Post by Cobord »

_jal wrote: post #49
Cobord wrote: post #40

Earlier quoted context omitted.

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…

But I would think the same applies to your algorithms teacher. There should be just as many good and bad ones there. Yet it seems different level of what is counted as too rote and unusable.

Re: Introduction to Differential Equations (2008)

#86

Post by ruang »

jshowa1 wrote: post #71
craig_asp wrote: post #6

Earlier quoted context omitted.

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".

You need to learn the math before applying it. You don't apply things that you don't understand.

I need the concrete before the abstract just so my mind knows that what I’m seeing is not BS. Because in finance, 95% of the math is BS formulas that have no connection to reality.

Re: Introduction to Differential Equations (2008)

#87

Post by jshowa1 »

amelius wrote: post #82
jshowa1 wrote: post #81

Earlier quoted context omitted.

Laplace transformations are differential equations so I fail to see your point. They're just in a different domain. However I do see your point with numerical methods since most complex problems are simulated anyways through simulating software. So in essence, the application becomes pointless because its at such a higher level of abstraction that you don't even have to think about it. You just punch in some numbers…

> Laplace transformations are differential equations so I fail to see your point. What I mean is that typically an electrical engineer will convert L and C elements to complex impedances (which depend on the frequency through s), and will then compute as though the elements are ordinary resistances. The expression "d/dt" isn't used in the entire analysis. See: https://en.wikipedia.org/wiki/Phasor Quoting: > the phaso…

This is like saying that if I convert mph to m/s then it's not a speed anymore. It's still a differential equation, just in a different domain because you can convert back from the s-domain into the time one.

Re: Introduction to Differential Equations (2008)

#88

Post by jshowa1 »

freetime2 wrote: post #83
jshowa1 wrote: post #72

Earlier quoted context omitted.

"Such a teacher will be successful if at the end of the course every one of his or her students feels they have taken “a good course,” even though they may not quite be able to pin down anything specific they have learned in the course." Basically the equivalent of, if the course makes you feel good, then the course is good. Not something that I'd advocate. By the way, a differential equation is simply an equation wi…

Actually, I have degree in math from a reputable university. So while you are certainly entitled to think that isn't going "far enough in math", I do think that your opinion is in the minority here. If anything, I would say the fact that I have completely forgotten everything from that class is evidence I may have actually gone too far in math... or further than I ended up needing. And I think it's evidence that the…

I've used state machines on the job. In fact, entire architectures are designed around finite state machines. I've also solved a complex logic problem a senior engineer couldn't solve by implementing K-maps.

And of course my opinion is in the minority because nearly everybody under the sun complains about how useless college is and how things should be taught with more application without realizing that things are taught minus application for a reason (so that you can apply things generally instead of specifically) and that many of the hot technologies are just re-purposed PhD research.

Also, the effort many students give to college is less than average (at least from personal experience going through a private engineering school) and probably for most college students. So their complaints are really just the result of laziness and lack of responsibility more than anything.

A minority opinion does not make it invalid or worth less, unless you have evidence to discredit it.

Re: Introduction to Differential Equations (2008)

#89

Post by acjohnson55 »

jshowa1 wrote: post #80
acjohnson55 wrote: post #78

Earlier quoted context omitted.

There are certainly people who can learn that way, but it's not effective for most people. Most people have a limit to the amount of abstraction they can operate under before they need a tangible connection. Once that connection is made, most people can continue on to higher levels of abstraction.

The problem is math is entirely abstract. So you must learn the abstraction before applying it. Otherwise, you learn how to add 2 + 3, but you don't learn how to add n + m.

That strikes me as a strange way of looking at it. Most math taught to people who aren't pure mathematicians is taught precisely because of application to concrete situations. We value pure math largely because of the potential for future concrete applications.

In case you aren't aware (and I wasn't until I trained to be a teacher, so this isn't meant to be condescending), there are alternative methods of teaching besides abstraction-first. See https://en.m.wikipedia.org/wiki/Inquiry-based_learning

Re: Introduction to Differential Equations (2008)

#90

Post by freetime2 »

jshowa1 wrote: post #88
freetime2 wrote: post #83

Earlier quoted context omitted.

Actually, I have degree in math from a reputable university. So while you are certainly entitled to think that isn't going "far enough in math", I do think that your opinion is in the minority here. If anything, I would say the fact that I have completely forgotten everything from that class is evidence I may have actually gone too far in math... or further than I ended up needing. And I think it's evidence that the…

I've used state machines on the job. In fact, entire architectures are designed around finite state machines. I've also solved a complex logic problem a senior engineer couldn't solve by implementing K-maps. And of course my opinion is in the minority because nearly everybody under the sun complains about how useless college is and how things should be taught with more application without realizing that things are ta…

> I've used state machines on the job. In fact, entire architectures are designed around finite state machines.

My point wasn't that FSMs are useless. My point was that despite the fact that I personally have never needed to convert an NFA to a DFA in my professional career or program a turing machine, I still have a deep appreciation for those courses because they fundamentally changed the way I think about computation.

> I've also solved a complex logic problem a senior engineer couldn't solve by implementing K-maps.

While you are clearly very proud of this fact, I'm not sure why that's relevant here?

> everybody under the sun complains about how useless college is and how things should be taught with more application without realizing that things are taught minus application for a reason (so that you can apply things generally instead of specifically)

This is basically the exact opposite of my complaint. I was complaining that differential equation courses essentially focus on teaching a bag of tricks for solving specific types of equations. I'm sure that behind each of those tricks there is a very fascinating how and why that - upon deeper exploration - may have changed the way I think about numbers. But that certainly was not the focus of the class that I took.

> the effort many students give to college is less than average

So you're saying in a given population, many of its members will be less than average? Very insightful. If only I had gone further in math maybe I would be capable of such insights, too. :)

> So their complaints are really just the result of laziness and lack of responsibility more than anything.

Be careful with this line of thinking. You could say the same thing to discredit any attempt to improve the way a course is taught. But surely you must agree there is room for improvement, right?

> A minority opinion does not make it invalid or worth less, unless you have evidence to discredit it.

Given that you made no attempt to substantiate your opinion - it seems to me that the logical thing to do here is to side with the majority.

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