Upper Division ODE vs PDE class

I'm trying to decide between taking an ODE class or a PDE class next. I have already done Calculus 1,2,3 so I already know some ODEs and PDEs and linear algebra. I'm a 3rd year mathematics major with a minor in Statistics and I'm interested in applied mathematics.

ODE course coverage:

Ordinary Differential Equations
Existence, uniqueness, and stability; the geometry of phase space; linear systems and hyperbolicity; maps and diffeomorphisms.

Chaotic Dynamics and Bifurcation Theory
Hyperbolic structure and chaos; center manifolds; bifurcation theory; and the Feigenbaum and Ruelle-Takens cascades to strange attractors. Poincare-Bendixson theory.

PDE course coverage:

Method of characteristics, understanding derivations of canonical PDEs. Wave, heat, and potential equations.
Fourier series; Solve boundary value problems for heat and wave equations; Fourier transform; Lapace's equation; generalized functions; and numerical methods for approximating solutions of 2nd order PDEs.

I know that the difficulty of a course is loosely related to the course material.
So I was wondering which of them would be harder?
Which one of them would be more educational (i.e. I would learn more out of it)?
And which one would be more fun in your opinion?

Thanks.

AlephZero
Homework Helper
The ODE looks like a proper math course. The PDE looks like a fairly gentle tour around a few standard equations and numerical methods (but not necessarily methods that anybody actually uses in the 21st century!)

I would say the ODE course is harder and you would learn more, and what you will learn applies to PDEs as well as ODEs. But which course s "more fun" depends what you think is fun.

Student100
Gold Member
If you liked calc at all, then ODE is super fun. I'm not a very mathish person, but I enjoyed ODE quite a bit.

I haven't done any PDE courses, so I can't compare it to that, this is purely from what I thought about ODE.

verty
Homework Helper
If you liked calc at all, then ODE is super fun. I'm not a very mathish person, but I enjoyed ODE quite a bit.

I haven't done any PDE courses, so I can't compare it to that, this is purely from what I thought about ODE.
I can't see how an ODE course is fun, you learn to solve equations (or learn the theory of the equations so as to be better able to solve them). Unless you like solving math problems, which surely qualifies as "mathish", it won't be enjoyable.

WannabeNewton
I can't see how an ODE course is fun, you learn to solve equations (or learn the theory of the equations so as to be better able to solve them). Unless you like solving math problems, which surely qualifies as "mathish", it won't be enjoyable.
I agree with this sentiment that engineering/physics ODE classes (i.e. cookbook classes on solving ODEs) are the most boring things this side of the universe. Proper proof based ODE classes, ideally heavy in analysis, are a totally different story though. They can be extremely exciting and fun to take.

From a geometric point of view they are also very beautiful. Here's my most favorite book on ODEs: https://www.amazon.com/dp/0262510189/?tag=pfamazon01-20&tag=pfamazon01-20

The ODE looks like a proper math course. The PDE looks like a fairly gentle tour around a few standard equations and numerical methods (but not necessarily methods that anybody actually uses in the 21st century!)

I would say the ODE course is harder and you would learn more, and what you will learn applies to PDEs as well as ODEs. But which course s "more fun" depends what you think is fun.
I found a more sophisticated outline of the second half of the PDE course:

Separation of variables - review
Fourier series - finding the coefficients
Full Fourier series
Orthogonality, generalized Fourier series
Convergence of Fourier series
Completeness
Inhomogeneous boundary value problems
Laplace's equation: properties
Harmonic functions in rectangles and cubes
Poisson's formula
Wedges, annuli, exterior of a circle
Green's first identity
Green's second identity, Green's functions
Green's functions for the half-space and sphere (See section 7.4 in Strauss)
The Fourier transform and source functions (See sections 12.3, 12.4 in Strauss)
Computation of solutions: introduction
Finite differences for the heat equation
Finite differences for the wave equation

I agree with this sentiment that engineering/physics ODE classes (i.e. cookbook classes on solving ODEs) are the most boring things this side of the universe. Proper proof based ODE classes, ideally heavy in analysis, are a totally different story though. They can be extremely exciting and fun to take.

From a geometric point of view they are also very beautiful. Here's my most favorite book on ODEs: https://www.amazon.com/dp/0262510189/?tag=pfamazon01-20&tag=pfamazon01-20
Interesting, I don't really think my class is an engineering/physics ODE class that is structured like the Calculus sequences taken by all majors (math, physics, chemists and engineers). The exams contain both computation and conceptual questions.

I have managed to find some past exams for the first half of the ODE course:
http://birnir.math.ucsb.edu/files/bjorn/class-documents/119A_13_midterm.pdf

http://birnir.math.ucsb.edu/files/bjorn/class-documents/119A_13_final.pdf

And here are past exams from the PDE course:

http://www.math.ucsb.edu/~ponce/124A-f-08.pdf
http://www.math.ucsb.edu/~grigoryan/124B/exams/examM.pdf

They were not taught my the same professors, so the difficulty and course structure could be a bit different this year.

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I agree with this sentiment that engineering/physics ODE classes (i.e. cookbook classes on solving ODEs) are the most boring things this side of the universe. Proper proof based ODE classes, ideally heavy in analysis, are a totally different story though. They can be extremely exciting and fun to take.

From a geometric point of view they are also very beautiful. Here's my most favorite book on ODEs: https://www.amazon.com/dp/0262510189/?tag=pfamazon01-20&tag=pfamazon01-20
I really don't like that attitude towards engineering coursework that a lot of math and science majors take. They are not cookbook classes and if you don't understand the concepts and theory then you won't be able to solve the problems, when I took the differential equations course for engineers we learned the theory and how the material actually applies to real life systems. For example, there were mixing problems which are important for a lot of industries as simple as they may seem. We also had more complicated problems that involved more complicated systems, but nonetheless they were real world applications. Mind you I have also taken the differential equations course for math majors and it was honestly boring, the same theory as the engineering courses in more detail of course but those classes often leave you wondering why am I learning this material. Physics majors take courses in fluid mechanics, but I assure you a good bit of them have no idea what the material is useful for.

Now to answer the question, I was forced to take an applied mathematics for engineers course as part of our nuclear engineering curriculum. The course is essentially a PDE course, and I have to say for nuclear engineering we don't use any of the material from that course except for eigenvalues which I picked up in linear algebra. A lot of the material is as another poster mentioned outdated and is usually the round about way to do things, especially when you get to the wave and heat equations. For problems that would take me maybe a sheet to solve using concepts and techniques learned in my heat transfer course, it took me multiple pages to solve the same type problem using techniques from the applied mathematics course. Not only that but the problems were boring and the book did a poor job of explaining concepts, this is the book we used if you are interested https://www.amazon.com/dp/1439816247/?tag=pfamazon01-20&tag=pfamazon01-20

Just wondering, what would you guys say is the "harder" course, ODE's or PDE's? I'm almost done with my ODE class and I found it fairly easy so I'm just wondering.

ODEs definitely

The ODE course will be almost completely useless for your interests in applied mathematics, from what I can tell. I am simultaneously taking both courses you mention. The ODE's course is "interesting" and I've learned a few useful things, but practically everything I have learned in the PDE course is useful and relevant to my interests (in physics, granted, but that is essentially applied mathematics anyway).

In discussions with applied mathematicians working on problems such as numerical neuroscience, mathematical physics, fluid mechanics etc, knowledge of numerical methods significantly trumps knowledge of pure mathematical theory, to the point where the theory is a nice background, but not terribly useful for research (anybody who would like to object this point is more than welcome, I'm rather interested in seeing a good justification for pure mathematics).

It is also incorrect to suggest that the pure ODE course will be harder or easier, this is entirely dependent upon the professor.

WannabeNewton
(anybody who would like to object this point is more than welcome, I'm rather interested in seeing a good justification for pure mathematics).
I'm not sure that you phrased this statement correctly because as it stands it is a hilariously ridiculous statement: the "justification" for pure mathematics is self-evident, even in the context that you specified.

Student100
Gold Member
In discussions with applied mathematicians working on problems such as numerical neuroscience, mathematical physics, fluid mechanics etc, knowledge of numerical methods significantly trumps knowledge of pure mathematical theory, to the point where the theory is a nice background, but not terribly useful for research (anybody who would like to object this point is more than welcome, I'm rather interested in seeing a good justification for pure mathematics.
That almost seems like suggesting physics is worthless because you could just do engineering classes.

AlephZero
Homework Helper
In discussions with applied mathematicians working on problems such as numerical neuroscience, mathematical physics, fluid mechanics etc, knowledge of numerical methods significantly trumps knowledge of pure mathematical theory, to the point where the theory is a nice background, but not terribly useful for research (anybody who would like to object this point is more than welcome, I'm rather interested in seeing a good justification for pure mathematics).
That's not quite right IMO.

Knowledge of the math theory plus zero knowledge of numerical methods is completely useless, unless you plan to do all your work writing on parchment with a quill pen.

Knowledge of numerical methods plus zero knowledge of the math theory is not so much useless as dangerous, especially if you try to write your own software. (But I've seem so many examples of this over the last 20 or 30 years that it seems to be the "default" situation at grad student level).

Knowledge of both is ... well, not useless, and not so dangerous as the above (nothing is 100% safe, of course). And sometimes, it even stops you wasting a few months or years trying to compute the uncomputable.

I mean, aerospace engineers happily carry on without knowing for sure if there are unique solutions for various boundary conditions of the Navier Stokes equation, or even if there are solutions in general; does the Navier-Stokes problem pose a genuine difficulty for the engineers which requires that the mathematicians ride to the rescue? Again, I claim no expertise here; if I'm completely wrong and proving existence and uniqueness theorems for Navier Stokes would have some meaningful impact on the applications, I'd be happy to hear it.

I've posed the question to several applied and pure mathematicians more generally, and they usually stumble to find any insight from the study of pure differential equations which has proven useful in applications.

To answer your question student100, physics theory often precedes engineering innovations; physicists made the first break throughs with the transistor using modern physical theory which had no previous applications.

Pure mathematics has made no contribution to modern physics since the 19th century, from what I can see. Hint: String theory and wormholes (i.e. experimentally unverified conjectures) are not valid examples.

But I am not trying to start a flame war here; I am merely suggesting to the OP that interests in applied mathematics are better served by, well, applied mathematics courses.

That's not quite right IMO.

Knowledge of numerical methods plus zero knowledge of the math theory is not so much useless as dangerous, especially if you try to write your own software. (But I've seem so many examples of this over the last 20 or 30 years that it seems to be the "default" situation at grad student level).
.
This is really interesting, I've yet to hear this; could you elaborate? I actually have long thought pure mathematics to be quite beautiful and have enjoyed many such courses, and (from a personal standpoint) would like to see a reason to think that it's not just all fluff.

I mean, aerospace engineers happily carry on without knowing for sure if there are unique solutions for various boundary conditions of the Navier Stokes equation, or even if there are solutions in general; does the Navier-Stokes problem pose a genuine difficulty for the engineers which requires that the mathematicians ride to the rescue? Again, I claim no expertise here; if I'm completely wrong and proving existence and uniqueness theorems for Navier Stokes would have some meaningful impact on the applications, I'd be happy to hear it.

I've posed the question to several applied and pure mathematicians more generally, and they usually stumble to find any insight from the study of pure differential equations which has proven useful in applications.

To answer your question student100, physics theory often precedes engineering innovations; physicists made the first break throughs with the transistor using modern physical theory which had no previous applications.

Pure mathematics has made no contribution to modern physics since the 19th century, from what I can see. Hint: String theory and wormholes (i.e. experimentally unverified conjectures) are not valid examples.

But I am not trying to start a flame war here; I am merely suggesting to the OP that interests in applied mathematics are better served by, well, applied mathematics courses.
Group theory is used all the time in chemistry: http://chemwiki.ucdavis.edu/Physical_Chemistry/Symmetry/Group_Theory%3A_Application [Broken]
Differential geometry and topology is used in General Relativity, see books like Wald.
Harmonic analysis can be used in image processing and give rise to stuff like wavelets.
Number theory is used in computer security.
And above all, it is philosophically pleasing to put all the physics on a rigorous footing.

And can you tell me how useful QFT, QED or the entire quest for the Higgs boson is? How useful is the study of cosmology and the big bang? I guess it's all fluff too.

I think you shouldn't be so dismissive of what other people do.

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pasmith
Homework Helper
I mean, aerospace engineers happily carry on without knowing for sure if there are unique solutions for various boundary conditions of the Navier Stokes equation, or even if there are solutions in general; does the Navier-Stokes problem pose a genuine difficulty for the engineers which requires that the mathematicians ride to the rescue?
During a undergraduate mathematics course on fluid dynamics at Cambridge in about 2003, the lecturer made a statement to the effect that aerodynamics was a boring, solved problem. "Planes fly, what else do you want to know?"

Some years later, I read a paper in the Journal of Fluid Mechanics by an engineer at Boeing who was not satisfied with the conventional explanation for why planes fly (the first sentence of the paper runs "This work was motivated by long-standing dissatisfaction with the theory of induced drag, in quite a few respects.")

To answer the OP's question: the ODE course will at least expose you to analysis of non-linear systems. All the equations discussed in the PDE course are linear.

Well, this is an interesting subject, but rather than derail this thread, I'll make a new one.

WannabeNewton
You seem to be speaking off the cuff without knowing much if any of the uses of pure math is what this seems like. The "justification" for pure math isn't gauged by how it fits into engineering. Why the hell would pure mathematicians care about that? If that's the methodology then heck a lot of esoteric subsets of physics are "unjustified".

Why do the conclusions of pure mathematics seem so nonsensical when compared to the real world, then? For instance, I've heard it told that "matrices with repeated eigenvalues are not generic; they are, in a sense, rare." This observation, that there are far more ways for a characteristic polynomial to not touch the axis than there are ways for it to touch the axis, seems utterly meaningless (if it is not meaningless, be my guest to point out why) when one is confronted with the fact that repeated eigenvalues occur very routinely in applications. Another professor declared that "separation of variables is a terrible way to solve PDE's"; but an applied mathematician later told me that it is a trick they always try if they can.

Notice how whenever new physics is passed down to the engineers, it's not really simplified. If you're going to do research in solid state electronics, you need a thorough background in quantum mechanics, as good as any physicist working in the same field (from what I know). Yet nobody except mathematicians cares about the extreme rigor imposed in say, real analysis or advanced calculus (None of the theoretical physicists I've talked to have ever even taken such courses).

To me, math is a symbolic language used to describe problems. An algebraic equation solved for a variable is a organizational, book keeping tool to keep the information organized; you could just think it through without using the visual organization, but this would be slower and more error prone. The same holds for integrals, differential equations, and all manner of other computational things. Studying such a tool in and of itself is odd to begin with, but it's odder still to study the hammer with a disdain for hitting nails, and a belief that hammers are interesting by themselves.

This is why I suggest to the OP that s/he concentrate more on applied courses in computational methods and applied mathematics, as they will be much more useful to him/her.

jgens
Gold Member
Why do the conclusions of pure mathematics seem so nonsensical when compared to the real world, then?
The short answer is that you are mistaken here. The slightly longer answer is that you probably have no honest idea what types of questions are investigated in pure mathematics.

For instance, I've heard it told that "matrices with repeated eigenvalues are not generic; they are, in a sense, rare." This observation, that there are far more ways for a characteristic polynomial to not touch the axis than there are ways for it to touch the axis, seems utterly meaningless (if it is not meaningless, be my guest to point out why) when one is confronted with the fact that repeated eigenvalues occur very routinely in applications.
The result you mention can be trivially verified in some sense and is not something of genuine interest to mathematicians. It really makes little sense as an example to be honest. In any case, matrices with repeated eigenvalues have been studied extensively in mathematics. Lastly if you doubt whether matrices without repeated eigenvalues have any applications in physics or engineering, then you probably know very little of either in the first place.

Another professor declared that "separation of variables is a terrible way to solve PDE's"; but an applied mathematician later told me that it is a trick they always try if they can.
Citing an isolated opinion hardly makes your case. I also interpret the claim a little differently. I would guess the professor meant something like: "Separation of variables only works in very rare cases, so from that standpoint it is not a great way to solve PDEs. But if you happen to come across one of those rare PDEs for which it works, then use it."

Yet nobody except mathematicians cares about the extreme rigor imposed in say, real analysis or advanced calculus (None of the theoretical physicists I've talked to have ever even taken such courses).
Representation theory is pretty important quantum physics. For this you are going to need a Haar measure and (aside from some trivial examples) you actually need some real/functional analysis to get one. I can also name plenty of theoretical physicists and chemists at my school who have taken such courses. So there's that.

To me, math is a symbolic language used to describe problems. An algebraic equation solved for a variable is a organizational, book keeping tool to keep the information organized; you could just think it through without using the visual organization, but this would be slower and more error prone. The same holds for integrals, differential equations, and all manner of other computational things. Studying such a tool in and of itself is odd to begin with, but it's odder still to study the hammer with a disdain for hitting nails, and a belief that hammers are interesting by themselves.
Unlike the previous paragraphs, this view is not demonstrably wrong, but it is misguided. Much of modern mathematics is guided by aesthetics and as a result lots of things studied are actually pretty interesting. Maybe not from an applications to physics standpoint (although you should check stuff like this out: http://ncatlab.org/nlab/show/higher+category+theory+and+physics) but there are absolutely other reasons that a subject can be worth studying. Also most mathematicians--at least the ones here at Chicago--do not have this "disdain" for applications. Honestly I think that perception comes from a bunch of undergrad math and physics students who do not know any better.

I concede that statements of the form "A professor said x" or "Most people of category A I have met say/do/believe B" are not helpful, so instead I would like to focus on the following:

Representation theory is pretty important quantum physics. For this you are going to need a Haar measure and (aside from some trivial examples) you actually need some real/functional analysis to get one. I can also name plenty of theoretical physicists and chemists at my school who have taken such courses. So there's that.
I have seen representation theory used to produce neatly packaged derivations of say, the Dirac equation, but what I would like to know (and I confess no deep expertise in quantum field theory) what use has it been put to in the prediction of new and useful results? Clearly if it is highly useful in modern theoretical physics, I will have to eat my words.

I am merely suggesting to the OP that interests in applied mathematics are better served by, well, applied mathematics courses.
Makes sense to me. People who love the "purity" of their subject will have a different set of goals they hope to get out of their courses than those who are interested in un-pure applications. As such, classes are often tailored for one or the other depending on the interest. I never took an upper division ODE class, but I did study un-pure PDEs in applied classes and I thought it was fun. Fourier stuff, the common physics equations, complex numbers. Also, the combining and using of things you have just studied is fun and how they all actually come together into something you can actually see doing in some work setting is cool. Personally, I would rather continue with study into solving these equations numerically on a computer rather than explore the theorems and subtitles of the axioms. Clearly the immediate economic need does not lie toward the latter, and accordingly more students should and do study the former.

jgens
Gold Member
I have seen representation theory used to produce neatly packaged derivations of say, the Dirac equation, but what I would like to know (and I confess no deep expertise in quantum field theory) what use has it been put to in the prediction of new and useful results? Clearly if it is highly useful in modern theoretical physics, I will have to eat my words.
No doubt things like representation theory provide enormous computational tools for physicists in QM and QFT. As to whether it has predicted new/useful results or just tidied things up after the fact is outside my knowledge. Having only learned modern-dress formulations of the material I am generally clueless about the historical development of the subject. I am under the impression however that many results in particle physics were predicted on the basis of group/representation theory well before they were actually observed. This Wikipedia article might help: http://en.wikipedia.org/wiki/Particle_physics_and_representation_theory

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