Learning Higher Math

OK, so scientific principles that I do not understand really annoy me. Higher maths (post-calculus), having seemingly little resemblance to any other sort of math, fall into this category for me so I have taken it upon myself to learn more. I have gotten about all I am going to get from the web (which wasn’t much) and am looking for a more detailed look into the subject.

So, first, what broad areas are out there? I need to know what I don’t know. I have a doctorate in engineering, so have a good understanding of everything up through and including calculus. Right now it looks like the next areas to explore would be Number Theory and Set Theory. What else am I missing?

What would be a good basic book[s] on the subject[s]? This looks like a lot of voodoo to me, so I need something that starts from the beginning, but have no fear of lots of equations or things getting technical quickly. I was thinking a college textbook might be good, but I would prefer not to spend a couple hundred dollars on one. My goal is to understand the ideas behind the field, be conversant in the underlying theories, and be capable of reading a current paper in the field without immediately throwing up my hands and giving up.

Also, let me know if my goals are unrealistic. Thanks.

Have you studied Discrete Mathematics? That’s a precursor to a lot of Computer Science topics as well as some Linguistics ones. This is the field to which some classic puzzles such as the Towers of Hanoi belong. CS people are keen on this field of math.

PS Set Theory is a part of Discrete.

Linear Algebra, Fourier Transforms, Infinite Series, Vector Calculus…

We had a class that used Mathematical Methods for the Physical Sciences by Boas that was excellent. It condensed about 2-3 years of post-calculus math into 1 year, making it very user friendly. Learned all kinds of important stuff to understand advanced Physics. Get that book and do it from cover to cover.

I would recommend learning some statistics. It’s likely to be useful no matter what field you pursue, and maybe even in your day to day life. Given your mathematical background I recommend Hogg’s “Introduction to Mathematical Statistics”.

I’ll heartily second a recommendation for Boas (I don’t know how I would have made it through undergrad without that book), but I’m not sure it’s really what the OP is asking for. Most of what’s in that book is basically just fancy applications of calculus.

robert_columbia, number theory is certainly discrete, but I’d have a hard time saying that set theory is part of discrete math. In fact, most of mathematics nowadays seems to be based on set theory, so if anything, it’d be the other way around.

And to the OP, I’d be wary of thinking of anything as “higher math”, or “post-calculus”. Just because a subject is usually taught chronologically after calculus doesn’t necessarily mean that it’s based on calculus, or is any more difficult. In fact, many of the concepts in, say, number theory are first presented (albeit not in great depth) in elementary school.

Assuming that “calculus” above includes multi-variable calculus (and if it doesn’t, that would be a great subject for you to study), I’d suggest going through a text on linear algebra (as recommended by John Mace). Once you’ve done that, you’ll be ready to learn about abstract algebra, which is a gateway to a huge number of other subjects (particularly number theory, which you mentioned above, and topology, which you can learn partially without abstract algebra but is so much cooler with it.)

I’d recommend Calculus of Variations as a good next step up in complexity from “standard” calculus. Past that the walls between linear algebra and calculus really break down so good ability in both is required.

It’s the sort of stuff you would never learn in Calculus I, though. It does have a lot of applied stuff that I learned in Calculus II, IIRC.

I think you’re much younger than me, and it’s nice to know that book is still used. Or was used when you were in school. Most useful math course I ever took!

The core of an undergraduate math curriculum usually looks something like this:
[ul]
[li]Calculus[/li][li]Differential equations[/li][li]Linear algebra[/li][li]Real analysis[/li][li]Complex analysis[/li][li]Probability[/li][li]Abstract algebra[/li][/ul]
There are usually a few advanced electives offered, which may include number theory or set theory/logic. After that, people start taking master’s level classes, which generally consist of year-long second classes in real analysis, abstract algebra and an introduction to topology.

The focus in these classes is not so much on teaching you the material–although that is important–as it is on teaching you how to think like a mathematician. As an engineer, your main focus is on doing computations and getting the right answer, without too much worry as to why things work. As a mathematician, you’re trained to spend all of your time and mental energy on that last bit. This is a very difficult transition even for people who have instruction, and while it’s not impossible to get the mindset down on your own, I don’t like the odds of an autodidact pulling it off.

ETA: Terry Tao has a great essay on how a mathematician’s education works. Undergrad is basically getting you from stage one to stage two, and then grad school is all about getting you up to stage three.

You can also learn abstract algebra before linear algebra, if you like*. Or number theory before abstract algebra. Or computability theory before…

Take Chronos’s comment to heart: very little in math has to be taught in the order it’s conventionally taught in (e.g., if I had my way, everyone would study linear algebra before calculus; why study the complications of local linearity before linearity simpliciter?**). So instead of thinking “What’s the next area?”, think “What is it that I’m interested in ultimately?”. Then figure out the best way to get there.

You mentioned “scientific principles that I do not understand really annoy me”; does this mean your interest in mathematics is specifically in order to cast light on the physical sciences, rather than for its own sake? Because, if so, number theory seems an odd subject to pick (as is much of what is called set theory, though the basics of structural set theory are useful throughout mathematics).

If your interest really is in mathematics as a tool for science, my personal recommendation would be to study linear algebra (and by this I don’t mean mere matrix manipulation) till you are completely comfortable with it, then study or revisit multivariable calculus, probability, etc., buoyed by the insights and perspective you’ve gained from linear algebra.

But, of course, if anything else comes along on the way that you think you’d rather study, do that instead!

[*: You could think peacefully about plain abelian groups before worrying about abelian groups acted on by a fixed field…]

[** (rant expanded): In particular, I don’t think anyone really understands multivariable calculus before understanding linear algebra, and I would also say, only slightly facetiously, that there no such thing as single-variable calculus [consider that multiplication, exponentiation, etc., are all already functions of multiple variables; what is the “product rule” but the multivariable chain rule applied to the multivariable multiplication function? Yet ask calculus students to differentiate such a basic analogue as x^x, and watch their faces fill with confusion! (That or think of it as some kind of ad hoc logarithm trick…). But no wonder, since their whole education, they’ve been intentionally isolated from such insight…]]

ultrafilter’s post is a good one.

A “good understanding of everything up through and including calculus” could mean different things, including:

  1. a familiarity with the rules of calculus, algebra, etc. and how to use them to “do math” such as solving equations, finding derivatives, etc.
  2. an understanding of the theoretical basis for the math: a familiarity with the definitions, theorems, and proofs of calculus.
  3. an intuitive grasp of the ideas of calculus, etc.
  4. an ability to use math (calculus, etc.) as a problem-solving tool, for modeling real-life situations and solving problems.

Which of these do you have? Some math classes/curricula focus on some of these while neglecting others. And some “higher mathematics” is very abstract and theoretical (which means that to understand it, you have to be familiar with logical, mathematical reasoning, common proof techniques, etc.), and some is applied math (which means you have to understand not only the mathematics itself but the field it’s being applied to—which you might have a head start on).

To get a broad overview of “what’s out there,” you may want to look at some of the books I and others recommended in the thread Recommend a book on mathematics.

Yeah, it might be unrealistic to go from “this looks like a lot of voodoo to me” directly to “be capable of reading a current paper in the field.”

I’d ask the OP where do you want to end up? Is your goal to be able to understand the mathematics in relativity theory? Wiles proof? Some advanced math that you’ve seen in your engineering work? What? I think if you’re going to undertake a journey into higher math it’s good to have a goal in mind. If your goal is to be able to read a paper on number theory in a math journal, then, yes, start learning some number theory. My advice has always been to first pick up an elementary book, start reading and working through it (do the exercises!). If you get stuck, back up and pick another book or reference and get over the sticky part, then move on until you’re through the subject. My experience, however, tells me that it will be very difficult to get through the most advanced material by yourself. That’s why in grad school there are seminars, colloquia, class discussions, office hours, coffee rooms, etc. to hash things out with other like-minded people. Calculus is quite distant from the math that’s in the journals.

I think what people are telling you is that you need to flesh out which “scientific principles” you are aiming to master. Otherwise, we’re all just taking shots in the dark. It seems like you aren’t into math for math’s sake, but we need more than you’ve posted in order to give advice on the next step.

My thoughts entirely. Arguably most of the last two years of my Pure Mathematics degree were based on set theory, a concept I learned when I was about twelve. In fact the only “calculus” I did as my degree was in the first year when they made us have a more rounded Maths education before we decided to specialise. I found out that, in general, I am poor at it and as such struggled with Applied Mathematics, so went the Pure (set theory, number theory, group theory, Topology, Logic …) instead.

As an aside, I’ve always found it weird that Americans (from my experience of popular culture and talking to real, live ones) seem to have their maths education at school split up in to different things. They do trig, calculus, algebra and whatnot. In the UK I just had a “maths” class. We did all those bits in it, but never broke it down and had specific lessons/homework in, say “algebra”.

Thank you for all the responses. It appears I was not terribly clear.

I currently have a solid understanding of all the math I need for my job and research (though I do occasionally need to refresh myself). This is purely from an interest in math. I understand that basic set theory can be taught very early (in fact, wasn’t the basis of the “new math” movement?), but it appears to someone who is not active in the field that current research is focusing more on this area than in calculus, which is why I referred to it as “higher” math. I guess I meant more the “fields in which new research is being done”, or more broadly, “fields where I don’t understand much at all.”

So, what do I understand now?

Yes to all. I was a generic “math tutor” as an undergrad and therefore effectively retook algebra, calculus (in its many forms), matrixes, and infinite series every semester. Infinite series in particular does not become something you intuitively grasp until the third time around.

I am solid through real analysis, have done a bit of complex analysis, with a basic grounding in probability (purely from personal exploration rather than classes).

I am well-versed in the calculus of variations and quite fond of it. (Despite understanding the basis of it, it still seems a bit magic to me.)

Actually, now that I look into it, I have mostly done matrix work in linear algebra. So, anything beyond that I do not think I have delved into (or at least do not associate it with linear algebra).

So, things that seem interesting at the moment from what everyone has said: more advanced linear algebra, abstract algebra, Set Theory, and Number Theory.

As to why, I have always enjoyed a wide variety of subjects, including math. I would like to have gone further in it in school, but the requirements of getting an Aerospace degree left little time for any elective courses, and I had pretty well plumbed the depth of the math courses that would be applicable to my work. I am learning for its own sake here and do not have a particular goal. I guess I would like to continue as far into the field as I can manage and can hold my interest.

I understand that classes and discussion would greatly help my efforts, but I mostly worked without them even in college (I was well known for my lackluster attendance). I have great ability to learn directly from books so long as the books are well written though.

I will look into the thread recommending books on mathematics as a place to start. Thanks again for the time you guys have spent on this.