Help me appreciate differential equations?

[dkotschessaa]

I'm a math major, which means I genuinely like mathematics. But I may be becoming a "pure math snob:"

http://img.spikedmath.com/comics/446-pure-math-vs-applied-math.png

When I'm working on differential equations, I'm finding it very tedious, like it's "not my kind of math." My brain says "This is engineering crap, I shouldn't be doing this."

Yet some other part of my brain is saying, "hey, the machinery behind this is really quite fascinating." Doing proofs on this is probably more interesting than doing problems, I suspect.

I feel like I'm missing something. I feel like old Diffy Q. and I were not properly introduced. I've read that basically the entire point of doing calculus is so that you can do Differential equations which model all sorts of behavior. That they are incredibly useful, that they can save the world, etc. etc.

But I have that non-practical math side that says "ok, big deal. Give it to the physicists and engineers!"

But seriously, what am I missing? Can I learn to love this subject, or should I just avoid it for the rest of my studies?

-Dave K

 

[SteamKing]

You've probably been slogging through linear DEs. Non-linear DEs still have much work to be done from a pure math standpoint (showing existence and/or uniqueness of solutions).
Partial DEs (classified as either parabolic, elliptical, or hyperbolic) also have many questions which remain unanswered by mathematicians.

 

[Vargo]

I came across this post awhile ago, and your question reminds me of it.

http://mathoverflow.net/questions/2...-undergraduates-who-want-to-become-pure-mathe

To quote the top answer, Differential equation is "content-rich". In Diffy Qs, there are many facts and they matter because they mean something real. Undergrads are usually turned off by the subject because it is too big and too difficult to grasp at a first look. Linear algebra, on the other hand, is a nice little subject you can learn in a semester or two. After that it is just another thing in your toolbox. Incidentally, where do you use most of those linear algebra results? Differential equations. In fact the study of differential equations incorporates sophisticated ideas/theorems from many other branches of math including abstract algebra, topology, geometry, and of course analysis (analysis should go without saying).

I think this was mentioned in the link, but Arnold has a good (though somewhat difficult and sophisticated) book on ODE.

 

[Vargo]

By the way, that picture is funny!

 

[micromass]

I agree that a first course in differential equations is likely to be very boring. Those classes usually present different forms of differential equations and give a recipe on how to solve them. All you got to do is recognize which type of differential equation you're dealing with and start calculating. All of these things aren't very interesting from a pure math point-of-view. In fact: I don't think anybody finds those classes interesting. Even physicists and engineers are not interested in the actual solving of the differential equation.

That said: differential equations really is a very exciting topic. The first cookbook-type class doesn't really do justice to how incredibly beautiful the subject could be. Sadly enough, you usually need to know the recipes and the boring stuff, before you can go into the more interesting stuff. It's like you can't do research in commutative algebra without knowing how to solve a quadratic equation (well, actually, you can. But I don't think it's a good idea).

Some interesting questions about differential equations could be: can we classify which differential equations can be solved by elementary functions and which not? This is studied in differential Galois theory. Which differential equations can be solved at all, and when is the solution unique? These questions are studied in analysis and functional analysis.
A nice generalization of differential equations are "flows" on manifolds. These are very useful things: the Poincarre conjecture was solved by using Ricci flows.

So it's certainly not the case that diffy eqs are boring and useless to pure math. It's just the the introductory course makes it seem boring. I find this very sad, but I don't think there's anything we can do about it.

 

[dkotschessaa]

I kind of sensed some of what some of you are saying, but I almost just needed to hear it. I also saw in my textbook some stuff on chaos theory, and remember thinking "whoa, that's here? " Of course we are not covering that in class. sigh.

But, to ask a stupid question, what is it that makes it such a huge field? I see at my school we have seminar series dedicated to them, (which I would like to try to sit through, though I'm sure it'd be over my head.) Is it because of the diversity of applications or is it because of the nature of the mathematics itself. (I suppose it could be both).

-Dave K

 

[romsofia]

Well, if you like pure maths then one subject that comes directly from differerntial equations is lie theory which starts from the exponential matrix because it's a solution to a DE. So, I guess you have that to look forward to as you can dive directly into that subject after DE (assuming you have done LA, and calculus).

 

[espen180]

Maybe the books by Vladimir Arnold are up your alley? Here is his book on ODEs.
http://books.google.no/books?id=JUoyqlW7PZgC&dq

 

[Vargo]

Why is it huge? The simplest explanation is that differential equations contain within them the secrets of the myriad phenomena of the physical world. There is no simple mathematical theory that will directly explain all these phenomena. You have to take each of these equations on their own and analyze it separately. There are many powerful ideas and tools that can be applied to wide classes of equations, but they don't explain everything, and they are not so powerful as to come even close to cracking the secrets of all equations. Anything we don't understand is a potential research opportunity.

To get started in the subject some good subjects are Lie theory, Calculus of Variations, anything titled ODE or PDE, harmonic analysis. Mathematical physics is also a good place for learning about PDE problems.

Some unsolved problems (the 2nd link is from 1989):
http://www.claymath.org/millennium/Navier-Stokes_Equations/
http://iopscience.iop.org/0036-0279/44/4/R05/pdf/RMS_44_4_R05.pdf

 

[dkotschessaa]

I came across this post awhile ago, and your question reminds me of it.

http://mathoverflow.net/questions/2...-undergraduates-who-want-to-become-pure-mathe

To quote the top answer, Differential equation is "content-rich". In Diffy Qs, there are many facts and they matter because they mean something real. Undergrads are usually turned off by the subject because it is too big and too difficult to grasp at a first look. Linear algebra, on the other hand, is a nice little subject you can learn in a semester or two. After that it is just another thing in your toolbox. Incidentally, where do you use most of those linear algebra results? Differential equations. In fact the study of differential equations incorporates sophisticated ideas/theorems from many other branches of math including abstract algebra, topology, geometry, and of course analysis (analysis should go without saying).

I think this was mentioned in the link, but Arnold has a good (though somewhat difficult and sophisticated) book on ODE.

I'll be doing L.A. next semester and I'm looking forward to it.

Does vector calculus utilize D.E?

Is there some kind of description or diagram that shows how all these classes and different branches of mathematics sort of fit together? Is that a weird thing to ask? All I have to go on is my "prerequisite flow chart," for my major.

Thanks for your reply.

-Dave K

 

[dkotschessaa]

I agree that a first course in differential equations is likely to be very boring. Those classes usually present different forms of differential equations and give a recipe on how to solve them. All you got to do is recognize which type of differential equation you're dealing with and start calculating. All of these things aren't very interesting from a pure math point-of-view. In fact: I don't think anybody finds those classes interesting. Even physicists and engineers are not interested in the actual solving of the differential equation.

That said: differential equations really is a very exciting topic. The first cookbook-type class doesn't really do justice to how incredibly beautiful the subject could be. Sadly enough, you usually need to know the recipes and the boring stuff, before you can go into the more interesting stuff. It's like you can't do research in commutative algebra without knowing how to solve a quadratic equation (well, actually, you can. But I don't think it's a good idea).

Some interesting questions about differential equations could be: can we classify which differential equations can be solved by elementary functions and which not? This is studied in differential Galois theory. Which differential equations can be solved at all, and when is the solution unique? These questions are studied in analysis and functional analysis.
A nice generalization of differential equations are "flows" on manifolds. These are very useful things: the Poincarre conjecture was solved by using Ricci flows.

So it's certainly not the case that diffy eqs are boring and useless to pure math. It's just the the introductory course makes it seem boring. I find this very sad, but I don't think there's anything we can do about it.

Yeah, I really don't blame the course or the instructor, who is actually fantastic, and quite entertaining. He is also sympathetic towards "Ok, here comes a really long and tedious equation - ready?"

I just wish I had a better appreciation of it in general. But you all are being a great help in this direction.

-Dave K

 

[dkotschessaa]

Why is it huge? The simplest explanation is that differential equations contain within them the secrets of the myriad phenomena of the physical world. There is no simple mathematical theory that will directly explain all these phenomena. You have to take each of these equations on their own and analyze it separately. There are many powerful ideas and tools that can be applied to wide classes of equations, but they don't explain everything, and they are not so powerful as to come even close to cracking the secrets of all equations. Anything we don't understand is a potential research opportunity.

To get started in the subject some good subjects are Lie theory, Calculus of Variations, anything titled ODE or PDE, harmonic analysis. Mathematical physics is also a good place for learning about PDE problems.

Some unsolved problems (the 2nd link is from 1989):
http://www.claymath.org/millennium/Navier-Stokes_Equations/
http://iopscience.iop.org/0036-0279/44/4/R05/pdf/RMS_44_4_R05.pdf

Thanks for this.

Did DE pretty much start with the invention of Calculus ala Leibniz/Newton or did it develop later? (if anyone knows) I haven't gotten quite that far in my math history yet...

-Dave K

 

[dkotschessaa]

I came across this post awhile ago, and your question reminds me of it.

http://mathoverflow.net/questions/2...-undergraduates-who-want-to-become-pure-mathe

Thought this quote was kind of funny: "However, I have been putting off taking a required ordinary differential equations course (colloquially referred to as 'calc 4', though this seems inappropriate) which will likely be very computational and designed to cater to the overpopulation of engineering students at my university."

My course definitely caters to engineers!

Another comment: I'm glad I didn't put off taking it. I figure if I want to eventually take analysis, I should be as well versed in all manner of calculus as possible and not lose touch with it. In this sense I do kind of see DE as "Calc 4".

I also am thinking I would like to take vector calculus before analysis. I'm wondering about the relationship between DE and Vector Calculus. Perhaps Vector is "Calc 5" in this way of thinking.

In general I want to make sure I am taking a course with calculus in it every semester, so that by the time I get to analysis I'm not going "Ah crap, I forgot how to integrate." I've met people in this position (not that extreme, but people who have kind of lost touch with calculus and are therefore struggling in analysis).

Anyway, does this approach seem sensible?


-Dave K

 

[Vargo]

1. Yes I'm pretty sure D.E. started with Newton+Leibniz, then continued with the Bernoulli brothers and took off from there.

2. Vector calculus is used for partial differential equations, but not too much the other way around.

3. Most undergrad courses are fairly self-contained, so you can just take a variety of them and see what you like. Asking your advisor these questions wouldn't hurt. Really the various topics in math are quite intertwined, but in order to organize them into courses, they are separated around different core ideas which are then presented as independent disciplines.

4. Many mathematicians would not acknowledge that there is any distinction between pure and applied mathematics. I have read and heard several great mathematicians say something along the lines that "There is no such thing as applied mathematics, only mathematics that is applied to solve problems". As long as the work is logically sound, where is the meaningful distinction?

 

[SophusLies]

Maybe the books by Vladimir Arnold are up your alley? Here is his book on ODEs.
http://books.google.no/books?id=JUoyqlW7PZgC&dq

That book is flat out beautiful. I was recommended that book 3 years ago when I started back to school and was blown away. It's a nice blend of intuition and rigor which is always good for a pure math type.