An Engineer's Path to Topology

[boneh3ad]

Hi. I did my undergraduate work in mechanical engineering and I am working on a PhD in fluid mechanics right now. I am interested in expanding my mathematical toolbox to include topology and am looking for some advice on where to start.

What subjects/topics should I cover as a prerequisite to taking an introductory topology course. I was already planning on differential geometry.

Thanks!

 

[Fredrik]

An introductory course on topology covers limits (of sequences, nets and functions) and continuity (of functions) in the context of metric and topological spaces. It doesn't require you to know any differential geometry. Differential geometry on the other hand requires you to know some topology (because terms like "second countable Hausdorff space" are part of the definition of "manifold").

I would say that introductory topology doesn't have any prerequisites, other than basic stuff about set theory, so that you can quickly figure out the answer when you find yourself wondering something like "is $f\big(\bigcup_{i\in I} E_i\big)=\bigcup_{i\in I}f(E_i)$?"

So if you want a head start, just open up the topology book and study the basic definitions and a few proofs. Try to write down a few proofs in your own words as if you're writing them for someone else. It takes a while to learn to do that well.

 

[lavinia]

Hi. I did my undergraduate work in mechanical engineering and I am working on a PhD in fluid mechanics right now. I am interested in expanding my mathematical toolbox to include topology and am looking for some advice on where to start.

What subjects/topics should I cover as a prerequisite to taking an introductory topology course. I was already planning on differential geometry.

Thanks!

There is a lot of topology in the study of fluid flows. I would lean that area of topology that focuses on dynamical systems. I know little about this stuff but a browse through a good fluid dynamics book should be a guide. I will look for one for you.

 

[boneh3ad]

Ah, interesting. I may just sit in on differential geometry in the spring then simply because "Topology I" isn't offered until the Fall. It can't hurt at any rate.

Thanks for the advice. I know a little about set theory already but not a ton. I haven't branched into pure mathematics much, as most engineers don't ever have the need. Unfortunately, I suffer from the desire to delve more deeply into problems and I know topology has been used as a pretty novel tool in my field in recent years.

There is a lot of topology in the study of fluid flows. I would lean that area of topology that focuses on dynamical systems. I know little about this stuff but a browse through a good fluid dynamics book should be a guide. I will look for one for you.

Of course I appreciate the advice, though I would be at least mildly surprised if you manage to suggest a book that I don't either own already or have at least a little familiarity with. Be that as it may, I am eager to see what you come up with. I am sure I probably have already had a few brushes with topology without actually knowing it as well.

 

[homeomorphic]

This could be a somewhat difficult question because to give the best answer, it would help to know both topology and fluids, and, aside from a casual acquaintance, I'm sort of ignorant of fluids.

V.I. Arnold has a book about topological methods in hydrodynamics, which could be on your reading list eventually, but it looks fairly advanced. I'd read it if I ever have time, but I'm afraid it would probably take me too far afield. Although, I think I looked at the contents and surprisingly saw some things that might be relevant for me, so maybe I will try to read it eventually, since Arnold's books are always great.

Anyway, topology is a big field. The starting point is point set topology. Munkres Topology is kind of the classic for that. That's what I used. I guess it has its drawbacks, but it worked well enough for me.

But, before point set, it helps to study real analysis if you haven't already. That's where a lot of the motivation comes from. So, ideally, you ought to do that first. Munkres, for example, doesn't have any formal prerequisites. But he says in the preface that if you haven't done real analysis and maybe some metric spaces, a lot of the motivation will be missing.

For differential geometry, I think you could begin with the geometry of curves and surfaces, for which you can find books that don't require much (or any) topology.

 

[lavinia]

Ah, interesting. I may just sit in on differential geometry in the spring then simply because "Topology I" isn't offered until the Fall. It can't hurt at any rate.

Thanks for the advice. I know a little about set theory already but not a ton. I haven't branched into pure mathematics much, as most engineers don't ever have the need. Unfortunately, I suffer from the desire to delve more deeply into problems and I know topology has been used as a pretty novel tool in my field in recent years.



Of course I appreciate the advice, though I would be at least mildly surprised if you manage to suggest a book that I don't either own already or have at least a little familiarity with. Be that as it may, I am eager to see what you come up with. I am sure I probably have already had a few brushes with topology without actually knowing it as well.

Differential geometry is tough going and maybe not a direct path for you. The qualitative theory of ordinary differential equations seems to me to be more direct. There are elementary books on this.

 

[boneh3ad]

I think it would be pretty sad if I didn't have a fairly reasonable grasp on ODEs already being halfway to my PhD... ;)

 

[lavinia]

I think it would be pretty sad if I didn't have a fairly reasonable grasp on ODEs already being halfway to my PhD... ;)

Ok. But the qualitative theory is profound and is reachable though elementary theory - e.g the Poincare Bendixon Theorem. But maybe you already know this?

 

[boneh3ad]

Ok. But the qualitative theory is profound and is reachable though elementary theory - e.g the Poincare Bendixon Theorem. But maybe you already know this?

I know a bit about the Poincare-Bendixson Theorem, though that has been a relatively recent acquisition. I haven't really used it, so its a bit hazy still, but it is one of those things I want to move into and understand more. It is pretty central to nonlinear dynamical systems, and given that my research area is in boundary-layer stability, it is basically nonlinear dynamics with a fluid mechanics bent. Good, fascinating stuff.

 

[lavinia]

I know a bit about the Poincare-Bendixson Theorem, though that has been a relatively recent acquisition. I haven't really used it, so its a bit hazy still, but it is one of those things I want to move into and understand more. It is pretty central to nonlinear dynamical systems, and given that my research area is in boundary-layer stability, it is basically nonlinear dynamics with a fluid mechanics bent. Good, fascinating stuff.

I have a paper somwhere that discusses braided and linked magnetic flairs on the sun. i guess this is fluid dynamics of charged particles. I will find the name for you. Braids and links are intuitive and have a lot of interesting topology in them.

 

[Sina]

You should really learn Real Analysis first, which covers point set topology. It is the foundations of all finite dimensional geometry. Marsden's Elementary Classical Analysis is good and covers most of the point set topology you will need for such studies (you may acquire the rest as you go on from books like Lee- Topological Manifolds etc). Sitting down just to study topology is not very efficient so composing it with real analysis or manifold theory should be more rewarding.

But you are talking about fluid dynamics so you may have to put your hands unders some infinite dimensional cases in which case, Functional Analysis + Infinite Dimensional Geometry might be useful.

If we strip ourselves off the dimensional concerns, smooth ergodic and morse theory might be quite useful to you which again is more meaningful after a course on real analysis and some knowledge of distriubtions+measures. you may then expand your knowledge to non-equilibrium topological dynamics etc

 

[boneh3ad]

Real analysis added to the list...

I will likely have to self-study that sometime in the next semester and a half. I have a tiny bit of experience in real analysis, but certainly not much.

 

[Fredrik]

Most people study some real analysis before they study topology, and therefore know some topology before they take a course in topology. But I don't think it's necessary to do it this way. In my opinion, it makes just as much sense to do it the other way round.

Regarding differential geometry and topology, I would say that while you need some topology to fully understand the definition of "manifold", it's unlikely that you will have to use any topology later in the course. So if you can live with not understanding some of the terms that go into the definition of "manifold", you can study differential geometry and be quite good at it without knowing topology.

I know nothing about fluid dynamics, so I have have no idea how useful it is there.

I am sure I probably have already had a few brushes with topology without actually knowing it as well.

The definition of "limit" (of a sequence or a function). Every statement you've heard that involves the terms "open set", "closed set", "compact set" or "connected set". Proving that two (or more) definitions of "continuous" are equivalent. Etc. I'm sure you've come across some of that in a course with "calculus" or "analysis" in the title.

 

[Studiot]

Two books of note for your purposes.

Introduction to metric and topological spaces

Sutherland

Oxford University Press

Differential Geometry an integrated approach

Nirmala Prakash

Tata Magraw Hill

Prakash has a good round up of much of the geek speak of modern maths including set theory and topology. This bridges the gap between elementrary stuff like Venn diagrams and much more advanced stuff.

I can also think of a mech engineer who found

An introduction to differential geometry

Willmore

Oxford University Press


To be very helpful when designing marine hull shapes and otherwise in classical theory of surfaces.
go well

 

[Sina]

Well most of the calculations in manifold theory do not make use of topology but many of the proofs does. You really need to know topology to do manifold theory. Otherwise I think it would be waste of time. Just study Marsden elementary classical analysis first 1-2 chapters and solve some of the questions.

 

[homeomorphic]

Most people study some real analysis before they study topology, and therefore know some topology before they take a course in topology. But I don't think it's necessary to do it this way. In my opinion, it makes just as much sense to do it the other way round.

My problem with this is that there are things like the topologist's idea of continuity that I think are a little bit unmotivated without a little real analysis. You don't need that much of it. Take something like continuity. A topologist's definition of continuity is that preimages of open sets are open. What does that have to do with continuity? I don't like to make leaps of faith like that too often. It helps if you have some experience dealing with the delta epsilon definition for R^n or metric spaces. Then, you can prove it's equivalent in that setting. There are various things like that.

It also depends on what approach you take. Some books/profs might be better suited to doing point set before analysis.

I remember when I learned the definition of a topology of a set. I was taking real analysis concurrently with topology, so I think we hadn't gotten to metric spaces or topology on R^n. What made it work was my topology prof gave us some motivation when he gave us the definition (which came from real analysis-type considerations). Otherwise, I might have found it a little too abstract and weird.

 

[Fredrik]

I agree that a little motivation is definitely useful, and maybe even necessary, but don't authors of topology books usually include some of that? For example, I suspect that all topology books prove that those definitions of continuity are equivalent right after (or right before) the definition you're talking about.

 

[boneh3ad]

Well I certainly appreciate all your help so far, everyone. Thanks! Certainly keep it coming if you have more advice!

 

[homeomorphic]

I can't recall how Munkres treated continuity. I think he eventually does prove it is equivalent to the epsilon-delta definition. It's in there somewhere. I remember being confronted with it, and I just had to ask my prof in office hours.

In any case, it's at least helpful to have prior experience with real analysis, although it might not be absolutely necessary.

 

[boneh3ad]

Oh the dreaded epsilon-delta definition of limits... I remember learning that freshman year of my undergraduate work and just being totally bewildered by how circular it seemed.

I bet it would be much more clear to me now nearly 7 years later.

 

[homeomorphic]

Oh the dreaded epsilon-delta definition of limits... I remember learning that freshman year of my undergraduate work and just being totally bewildered by how circular it seemed.

I bet it would be much more clear to me now nearly 7 years later.

When I say I had to ask my prof in office hours, I was already very comfortable with the delta-epsilon definition. I just remember having to ask him what the topologist's definition had to do with it. So, obviously, it helps to be very comfortable with these analysis concepts to fully understand what's going on.

Before you ask what the topologist's definition has to do with continuity, you might ask what the analyst's definition has to do with it. Either in terms of sequences for nice spaces or delta-epsilon.

I first saw the delta-epsilon thing in high school and it really didn't make much sense back then.