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For engineering (e. g. electrical engineering) and physics, Calculus is important. But for a future mathematician, is the classical approach to Calculus still important? What is normally taught, as a minimum, in most Universities worldwide?

Added the [calculus] and the [multivariable-calculus] tags.

Edit: Hoping it is useful, I transcribe three comments of mine (to this question):

  1. I had [have] in mind for instance Tom Apostol's books, although learning differentiation before integration. (in response to Qiaochu Yuan's "What is the classical approach to calculus?")
  2. Elementary Calculus, continuous functions, functions of several variables, partial differentiation, implicit-functions, vectors and vector fields, multiple integrals, infinite series, uniform convergence, power series, Fourier series and integrals, etc. (in response to a comment by Geoff Robinson).
  3. I had [have] in mind calculus for math students, although I am a retired electrical engineer. (in response to Andy's comment "Are you talking about what is usually taught to engineers and physicists, or also about a calculus curriculum for math majors? ")

I decided to make this question a CW (see this meta question).

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    Sadly, for many mathematicians, teaching calculus to future engineering and physics students is how you pay the bills.2011-08-17
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    @MartianInvader: So I was taught in my engineering course.2011-08-17
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    What is the classical approach to calculus?2011-08-17
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    I can't see doing complex variables without first doing calculus.2011-08-17
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    @Qiaochu Yuan: I had in mind for instance Tom Apostol's books, although learning differentiation before integration.2011-08-17
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    @Geoff Robinson: Elementary Calculus, continuous functions, functions of several variables, partial differentiation, implicit-functions, vectors and vector fields, multiple integrals, infinite series, uniform convergence, power series, Fourier series and integrals, etc.2011-08-17
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    A tangential remark: looking at the kind of math I need to know now, and also looking at real world applications, I would say that undergrad programs should emphasize linear algebra more heavily than they currently do, esp. compared to calculus.2011-08-17
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    @Américo: everything you list is extremely important, even in pure mathematics. How could it not be?2011-08-17
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    @Am\'erico: If that's what you mean, I agree with Qiaochu: everything there is important for mathematics, and ideally they should be taught with rigour. Whether that is happening, even in some very good mathematics departments, is another question.2011-08-17
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    @Qiaochu: I think so. The topics I listed are some of the chapters in Advanced Calculus by Angus Taylor, recommented in 1970 to future engineers in *Análise Infinitesimal* ("Infinitesimal Analysis") , in IST, Lisbon, Portugal.2011-08-17
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    To me it seems somewhat like a question what is the role of language in the modern culture? Sure journalists and the like need it, but for future artists is the classical approach to reading and writing still important?2011-08-17
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    I think elementary calculus acts as a good motivation for a diverse advanced mathematical topics - analysis, set theory, topology etc.2011-08-17
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    @Americo: I'm unsure about what you're asking in the second question. Are you talkiing about what is usually taught to engeneers and physicists, or also about a calculus curriculum for math majors?2011-08-17
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    @Andy: I had in mind calculus for math students, although I am a retired electrical engineer.2011-08-17
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    @John, I think a few years ago there was a recommendation by the AMS or MAA to place emphasis on linear algebra.2011-08-17
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    @Americo: good, I'll undelete my answer then. :)2011-08-17
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    @Geoff The answer evidently seems to be no. But writers here correspond to those who use analysis constantly (complex/functional analysis, differential equations, geometry, calculus of variations, number theory etc.) But theoretically someone in combinatorics or logic could seem to not need any knowledge of calculus. All in all it it possible to paint without knowing letters. But even somebody from combinatorics could meet from time to time functions to be studied. And besides the things one could need here like asymptotic series the knowledge of calculus ideology would be handy as well.2011-08-17
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    I am unsure if I should add the [calculus] tag.2011-08-17
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    @Andrew Perhaps combinatorics, but logic might depend on which branch you want to study. If you want to study fuzzy logic, you'll study fuzzy set theory at some point, and then you'll need some basic calculus (though perhaps "analysis" would be the better term here) at some point.2011-08-18
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    This may as well be relevant to the conversation: http://mathoverflow.net/questions/52708/why-should-one-still-teach-riemann-integration2011-08-18
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    I flaged this question. I want to make it a CW but I do not remember how.2011-08-18
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    @Américo: Only moderators can make questions into a CW on this site.2011-08-18
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    @Zhen: Many thanks for the information!2011-08-18
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    @Américo: For what it's worth, I do not agree with the comment (made by someone who is not really a user of this site, by the way) that your question was "quite poorly written". There was some ambiguity in it which you quickly clarified in the comments. But your edits are definitely an improvement; I would venture to call the new version "good".2011-08-18
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    @Pete: Thanks for your opinion.2011-08-18
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    @Américo: I agree with what Pete said, but I couldn't remain silent because I didn't like that judgment either. It was a good question with interesting reactions, so don't take this assessment from one person too much to heart (also there are 19 upvotes and no downvote, so don't worry). Also, I wanted to tell you for a long time already that I don't spot that many English errors in your posts (I'm not a native speaker but still), and I called the mathematical ones I saw to your attention (once?, twice?). So no need to apologize in your profile!2011-08-18
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    @Theo: Thank you too for your opinion.2011-08-18

3 Answers 3

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In a comment to his question, Américo has clarified that by "classical calculus" he means something relatively rigorous and theoretical, as for instance in Apostol's book (or Spivak's). I think the answer to the question was probably yes no matter what, but when restricted in this way it becomes a big booming YES.

The methods of rigorous calculus -- may I say elementary real analysis? it seems more specific -- are an indispensable part of the cultural knowledge of all mathematicians, pure and applied. Not all mathematicians will directly use this material in their work: I for one am a mathematician with relatively broad interests almost to a fault, but I have never written "by the Fundamental Theorem of Calculus" or "by the Mean Value Theorem" in any of my research papers. But nevertheless familiarity and even deep understanding of these basic ideas and themes permeates all of modern mathematics. For instance, as an arithmetic geometer the functions I differentiate are usually polynomials or rational functions, but the idea of differentiation is still there, in fact abstracted in the notion of derivations and modules of differentials. One of the most important concepts in algebraic / arithmetic geometry is smoothness, and although you could in principle try to swallow this as a piece of pure algebra, I say good luck with that if you have never taken multivariable calculus and understood the inverse and implicit function theorems.

Eschewing "classical" mathematics in favor of more modern, abstract or specialized topics is one of the biggest traps a bright young student of mathematics can fall into. (If you spend any time at a place like Harvard, as I did as a graduate student, you see undergraduates falling for this with distressing regularity, almost as if the floor outside your office was carpeted with banana peels.) The people who created the fancy modern machinery did so by virtue of their knowledge of classical stuff, and are responding to it in ways that are profound even if they are unfortunately not made explicit. Although I am very far from really knowing what I'm talking about here, my feeling is that the analogy to the fine arts is rather apt: abstract modern art is very much a response to classical, figurative, realistic (I was tempted to say "mimetic", so I had better end this digression soon!) art: if you decide to forego learning about perspective in favor of arranging black squares on a white canvas, you're severely missing the point.

The material of elementary real analysis -- and even freshman calculus -- is remarkably rich. I have taught more or less the same freshman calculus courses about a dozen times, and each time I find something new to think about, sometimes in resonance with my other mathematical thoughts of the moment but sometimes I just find that I have the chance to stop and think about something that never occurred to me before. Once for instance I was talking about computing volumes of solids of revolution and it occurred to me that I had never thought about proving in general that the method of shells will give the same answer as the method of washers. It was pretty good fun to do it, and I mentioned it to a couple of my colleagues and they had a similar reaction: "No, I never thought of that before, but it sounds like fun." There are thousands of little projects and discoveries like this in freshman calculus.

I confess though that it would be interesting to hear mathematicians talk about parts of calculus that they never liked and never had any use for. As for me, I really dread the part of the course where we do related rates problems and min / max problems. The former seems like an exercise whose only point is to exploit -- sometimes to the point of cruelty -- the shakiness of students' understanding of implicit differentiation, and the latter was sort of fun for me for the first ten problems but twenty years and thousands of min / max problems later I could hardly imagine something more tedious. (Moreover I am not that good at these problems. I had a couple of embarrassing failures as a graduate student, and ever since I look to make sure I can do the problems before I assign them, something I have stopped needing to do in most other undergraduate courses.)

Added: Let me be explicit that I am not answering the second part of the question, i.e., what is a minimum that is or should be taught. It goes hand in hand with the richness of these topics that if you tried to make a list of everything that it would be valuable for students to know, your (surely severely incomplete!) list would contain vastly more material than could be reasonably covered in the allotted courses. This is one subject where books which aim to be "comprehensive" come off as pretty daunting. For instance I own the first of Courant and John's two volumes on advanced calculus: it's more than six hundred pages! Is there anything in there which I am willing to point to as "dispensable"? Not much. (Not to mention that the second volume of their work comes in two parts, the second part of which is itself 954 pages long!) The challenge of teaching these courses lies in the fact that the potential landscape is almost infinite and virtually none of it manifestly unimportant, so you have to make hard choices about what not to do.

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    +1. I totally agree with you that "eschewing 'classical' mathematics in favor of more modern, abstract or specialized topics is one of the biggest traps a bright young student of mathematics can fall into." This is really a wonderful answer. How I wish I can vote twice.2011-08-17
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    Dear Pete: thank you for posting this answer. I was thinking of saying something along these lines, but thought someone else might do it (better than I would have). Let me add -- as someone who likes abstraction more than most -- that one of the things I have admired about people like Jacob Lurie (to pick just one example) is not simply their fearsome technical competence about $\infty$-categories and stacks -- but their wide-ranging and in-depth knowledge of the more classical ideas in their field.2011-08-17
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    @Akhil: thanks. Just to be sure: I am not proscribing abstraction or technical specialization in mathematics. On the contrary these trends of modern mathematics are not just inevitable but beneficial: specialization allows a much larger group of people to make useful contributions and makes it less about who is cleverest or quickest. It's just that (for the vast majority of people) the route to specialized knowledge passes through core, classical knowledge. It is those who think that the classical stuff is a waste of time who are (in my opinion, obviously) mistaken.2011-08-17
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    Not much I can add to that wonderful post,Pete.I WILL say that mathematicians sadly tend to downplay-or worse,eliminate-the applied aspects of calculus when teaching calculus "carefully". This is throwing the baby out with the bathwater to me. I personally LOVE Courant and John precisely because it does NOT do this.I highly recommend it to any talented undergraduate in honors calculus-they are jusifiably classics and can be profited from by anyone either learning or teaching calculus.(to be continued)2011-08-17
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    (Continued)Sadly,the story you tell about the students at places like Harvard is very true.Even more sadly is the fact that most of the time,the faculty at these places encourage this behavior because they're more interested in turning out teenage "genius" researchers that can awe the media and wealthy alumni then properly trained scientists.This is why we have 20 year old students doing research in non-associative algebra geometry,yet don't know what a net is and are mystified by "big Oh" notation.2011-08-17
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    Right on, and can be concisely put by the proverb, "The best is the enemy of the good."2011-08-17
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    Risking some hate I would like to disagree here. First and foremost I would like to note that, though it probably was not meant to, at least I, as an "undergraduates falling for this", found this post rather condescending and frankly somewhat rude. Realising the rest of my reading might be somewhat coloured by this I would also like to say that I, and I doubt I am alone in this, find that the methods of the "the fancy modern machinery" is what is interesting about mathematics while being somewhat disinterested in its historical origins as anything but just that, historical background.2011-08-17
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    @Tilo Mathematics is unlike other disciplines in that it builds vertically from what has come before in it.It's very hard to see how a student without a strong grasp of classical real analysis is going to be able to understand-truly understand-operator theory and the modern theory of differential equations, as well as metric space theory so essential to both.2011-08-17
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    @Tilo: I'm sorry you found my answer condescending and rude. That was certainly not my intent. The question asked for, in part, an opinion about university level mathematics education. I am, in part, a math teacher at the university level, and I am giving my "professional" opinion and relating my experiences. Advice by its nature has a take-it-or-leave-it quality: if you choose not to take it, I would be interested to hear from you later on to see how things worked out. (Let me also note that one of the people who agreed with me above is an actual Harvard undergraduate...)2011-08-17
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    @Mathemagician: I don't want to argue with you here, but I don't agree with your characterization of Harvard faculty. They mean well and largely do well in training undergraduates. It is just sort of part of the Harvard ethos that they trust you enough to climb back out of the pits you may fall into. And I agree with the end result but think that in most cases the less tempestuous route is more efficient and more pleasant.2011-08-17
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    @Pete: It may very well be the case that there are people who have the issues you describe, I just felt that I had to say that there are those of us who find that it is the very "abstractness" of "modern mathematics" which makes it appealing.2011-08-18
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    @Tilo: I've seen what Pete is referring to. And this is a problem not just in mathematics, but many other subjects as well. An analogy to physics might be a string theory student that isn't familiar with the Standard Model, or the Higgs mechanism or something like that. In mathematics, one frequently comes across people who misunderstand the nature of a subject because of its abstractness -- theorems that they thought were deep theorems in their abstract subject being mere dressed-up versions of basic theorems in a far more classical area.2011-08-18
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    @Tilo: I understand the appeal of abstraction very well. However, I know quite a few people who deal with abstraction without being able to give a single reasonable example (I'm in danger of this, too). Model categories, derived categories, topoi, whatever else your favorite beast at the moment, are definitely nice and useful. But learning about them without knowing about their *motivation* seems a bit pointless. It is impossible to deny that modern set theory and category theory have *concrete* origins. Ignoring that may well mean missing a significant aspect if not the *entire point*.2011-08-18
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    @Ryan: I didn't see your comment before I posted mine. I'll leave it anyway since we make similar points in a quite different way.2011-08-18
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    To add to what Theo is saying: the *mathematics story* has a plot. And if you jump into the story mid-stream, skipping the early part of it, you're in danger of misunderstanding or even *missing the plot entirely*. The results can be extremely disorienting, especially since the mathematics story isn't as viewer-friendly as a Hollywood movie. Theo: yes, as-is is quite good. This comment is perhaps overkill now!2011-08-18
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    @Ryan: I am not very familiar with physics (or non-formal science in general) so I would not want to comment there. I do not see though, why a general theorem would be any less deep just because a special case of it (which I assume what it is you are referring to when you say "... dressed-up version of basic theorems ...") is simple. Note, by the way, that I am not stating that "classical" subjects cannot be useful for understanding modern mathematics, I am just very sceptical as to whether they are as essential as many people, not least here, seem to suggest.2011-08-18
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    @Theo: I do not think it is unreasonable to consider (not suggested it reduces to or that it is **the** way to consider) mathematics as a study of formal structures. When viewed from this angle I do not see why the objects of study in "classical" subjects would be any more "motivated" than those in "modern" ones (apart from, possibly, their application in applied mathematics, to the extent that such a distinction is meaningful). Perhaps I should stop now though, as comments are not meant to be used as a discussion forum (I could not help to expound what I meant earlier though).2011-08-18
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    @Mathemagician: Dear Andrew, I would also say that (at least in my experience) the faculty at Harvard and MIT generally encourage students to learn the basics thoroughly. (It is also true that they often expect the students to do this on their own as a base-line.)2011-08-18
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    @Tilo: if one were to compare the mathematics profession to cake decoration, you could say (1) the fundamental role of cake decoration is to make decorative cakes that appeal to people, so that people can have the kinds of cakes they like or (2) cake decoration is the art of decorating cakes. Both are true in a sense but both kind of aren't the whole story. I think this discussion is like that, an exercise in different levels of partial context.2011-08-18
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    At the risk of dragging this thread off topic, here is a bizarre anecdote: a couple of years ago, we had a professor who seemed to contravene every principle here. Legend has it he taught category theory in "Math 1b," the non-honors calculus course intended for students outside the sciences; in the undergraduate topology course, he apparently (this from someone who took it) started with a solid overview of point-set topology, and then decided the time was ripe to begin etale cohomology! But, such unusual instances notwithstanding, I almost never see experienced mathematicians...2011-08-18
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    ...even those who work in the most high-powered abstraction for a living -- advise their undergraduate students to neglect the foundations, or to ignore the classical origins of the subject. Even books like Bourbaki do spend a while discussing history and motivation. (For what it's worth, I've also been told that the aforementioned professor is, aside from his, er, non-standard curricula, an excellent teacher.)2011-08-18
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    @Akhil Sadly,Akhil-I completely believe it.I hope some of the other faculty,when hearing from students terrorized in that non-math major course,took him aside and explained to him that having students of the quality of MIT is a priviledge that shouldn't be abused!2011-08-18
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    @Pete I know there are many fine teachers at the top tier schools.Indeed-I understand Shlomo Sternberg has "found religion" and become a superb teacher in his later years.But the pressure in such places to publish often forces the undergraduate acquisition of the basic skills to take a back seat in priorities. (to be continued)2011-08-18
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    @Pete cont. This is particularly true if the student is quite advanced and shows immediate research skills.As a result,the student usually ends up with weaknesses and/or holes in their training that are never addressed. It is true they may very well go on and become fine mathematicians regardless.But if they DON'T,one does have to wonder if a more conventional education would have made a difference.It troubles me-especially with America's growing educational failures.That's all.2011-08-18
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    @Mathemagician: again, I largely disagree with your comments about faculty at the top math departments. But more fundamentally, I don't see what they have to do with the question or with my answer to it. It seems like your opinions would be more appropriate for your blog than for focused Q&A sites like this one.2011-08-18
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    @Mathemagician: I realize that trying to make a point while following it up with a counterexample -- even combined with the assertion that said counterexample is the rare exception -- was a bit silly of me, and in fact I thought your portrait unduly negative. Anyway, I agree with Pete that our discussion is getting off topic (and I apologize for my irrelevant comments above that prompted this); feel free to send email if you'd like to continue.2011-08-18
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[I've decided to weigh in even though I am neither particularly experienced in mathematics or pedagogy. But, now feels like a good time to procrastinate from work...]

It is possible to learn reasonable chunks of 20th century mathematics without knowing what a derivative is. For instance, let's take abstract algebraic geometry: most of Grothendieck's theory (as developed in EGA/Hartshorne) requires several prerequisites (sheaf theory, homological algebra, general topology, commutative rings) but notably not calculus. An advanced undergraduate (who has studied all this and much more) once told me he did not know the definition of the exponential function. You might object that general topology grew out of foundational questions in analysis, which in turn grew out of calculus; however, one can define the requisite notions (topological spaces, continuity, connectedness) from first principles. There are in fact essentially no prerequisites for starting general topology, interpreted suitably. Similarly, abstract algebra can be studied from naive set theory, starting with the definition of a group.

Now many of the important results in algebraic geometry do rely on analytic methods; to pick one example, the Kodaira vanishing theorem can be phrased as a purely algebraic statement about smooth projective varieties over a field of characteristic zero. But the usual proof uses complex analytic methods (Hodge theory), and calculus is certainly a logical prerequisite for them. Nonetheless, let's say that you wanted to shun every part of algebraic geometry that depended on analysis; there is still plenty of interesting stuff to think about.

(Maybe Kodaira vanishing is not the best example: Deligne and Illusie apparently found a purely algebraic proof, but only several decades later.)

So does it still make sense to know calculus? I think the answer is a clear yes even if you fall into the hard-line category above. More generally, it helps to have an awareness of the historical context of ideas. Mathematics tends to be heavily cross-pollinated: ideas from one field fertilize another. Many of the greatest ideas in one field are inspired by those of other fields, even if in the final product (the polished version that appears in papers or textbooks). I was recently reading a paper on number theory that claimed to be inspired by an argument of Witten for the very non-number-theoretic Morse inequalities.

Here's an example: there is a notion (as Pete Clark mentions) of a derivation of an algebra: it's a map that behaves like ordinary differentiation does, i.e. satisfies the Leibniz rule. It's entirely possible to define a derivation abstractly and memorize the definition as such, without understanding where it came from -- of course, calculus -- and work with it. In fact, it is possible to treat any mathematical idea in this way -- as a purely self-contained, isolated concept. But most of us (certainly including myself) would instinctively recoil at this.

In general, when confronted with a set of axioms, one asks why they are there. Anyone can dream up a mathematical structure, but only some are interesting; those that are interesting usually are because the axioms are intended to model some idea. For instance, groups model symmetries or transformations of an object. If you are aware of this, then the idea of a group representation becomes more intuitive than if you think of a group exclusively via its literal definition as a set structured in a particular manner.

Mathematics, historically, has not proceeded from the general to the specific, but from the specific to the general. (And back to the specific, sometimes.) Projective and affine varieties came before schemes, $\mathbb{Z}[i]$ came before general Dedekind domains, and integration in euclidean spaces came before modern measure theory. "Categories" may be foundational material, but they were invented to better understand algebraic topology -- a well-established discipline by then. It is of course impossible to learn mathematics in historical order; there is not enough time in one's life, and often there are shortcuts one can take to understanding classical material with a better modern understanding. But if you want to understand and work with the axioms in modern mathematical structures, it seems only natural that you should have some awareness of how people decided to put them in. (In fact, in all the above examples, the axioms for the relevant modern structures (schemes, measures, Dedekind domains) are precisely those intended to model the essential features of the classical examples.)

In short: it is possible to treat mathematics as solely a game played with meaningless marks on paper, devoid of history and culture, in which case ignoring something like calculus is probably feasible, at least if you stick to the appropriate subfields. (Apologies to David Hilbert and Zev Chonoles: neither endorses this approach, but I have to pick on the quote.) But this seems to me neither a sound approach nor a satisfying one.

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    Dear Akhil, thank you for your answer.2011-08-18
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For the second part of your question, namely "What is normally taught, as a minimum, in most Universities worldwide?", I'll try to give my two cents.

I live in Italy and here we don't make any difference between analysis and calculus courses. In my university we have four mandatory courses in analysis, which span from differentiability in one variable to measure theory. I'll give a brief chronological order in which I was taught the main "calculus" curriculum. You should keep in mind that everything had solid theoretical basis (every course is proof oriented).

  • Analysis 1

Metric spaces, limits [along with a lot of other stuff, which filled up the course]

This course had very little syllabus mainly because in "metric spaces" we would cover "all" topological properties for metric spaces: openness,closedness,compactness and so on. In the "limit" category we would fit in limits in metric spaces (related with characterization of compactness) and real functions of real variables limits, series and sequences. It was actually a very broad course.

  • Analysis 2

Differentiability in one variable, integration in one variable, differentiability in multiple variables, integration in multiple variables [this one was more "calculus" oriented, since the "full" analysis would be given later with measure theory. The variables were real.]

I guess the standard calculus curriculum stops here (but I'm not sure). This is covered in the first two semesters in the first year at my school (and usually all around Italy). The two other courses are in the second year and are way more theoretical than these two; let me know if you think they are relevant, I'll add something about them too.

As for you first question, I'm in no postion to answer, since I'm still young and don't really know where I'm going with all this.

EDIT: to answer @3Sphere's comment I can only tell you that the books recommended for those classes were

  • P.M. Soardi, Analisi Matematica
  • Rudin, Principles of Mathematical Analysis
  • Gelbaum, Holmsted, Counterxamples in Analysis

(for the first course)

  • C. Maderna, Analisi Matematica II

(for the second one)

I believe that there won't be any translations for those two books because they are from an editor that does not publish abroad. I'm sorry but I don't know of any italian book you could find in english, I'm sorry. On the other hand, I think that the suggetion of Rudin is quite an indicator that the presentation is quite international anyway (I hope this is clear).

EDIT 2: as per Américo's comment I'll add some info about the other two mandatory courses in analysis

  • Analysis 3 Differential equations, sequences and series of functions, integration over curves, differential forms and integration of differential forms over curves, Dini's theorem and implicit functions.

The main part of the program was differential equations, so basically everything else we did we reconnected to those, e.g. series solutions for differential equations and exact differential equations.

  • Analysis 4

Lesbegue measure and Lesbegue integral, general measure theory, Haussdorf measure.

This was meant as a natural generalization of the multivariable part of Analysis 2.

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    I found that it is more common that the multivariable part of the classes I described be taught during the second year. My school has just (a couple of years ago) changed the curriculum.2011-08-17
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    What people generally call college calculus here is covered in HIGH SCHOOL in my ancestral land,Andy. You'll see similar cirricula in Germany and some parts of France as well.This is one big reason why America is slowly becoming a third world country academically as well as economically.2011-08-17
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    You can actually cover differential and integral calculus in one variable (with some theory behind it) in one "kind" of high school here (the system is quite different from the US). But still, when I spent my 4th year of high school in California (senior year for my class), I found the math curriculum wasn't that bad at all, when compared to my "kind" of high school in Italy, which was literature oriented [I didn't like math when I was little].2011-08-17
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    @Andy It's interesting to know what constitutes "Analysis" in other countries (meaning other than here in the US). Is there a sequence of (English language) texts that you could recommend that would be reflective of the Analysis syllabus you describe? I have seen the analysis texts of Canuto and Tabacco, which are Italian texts that have been translated to English, but their content seems more in-line with computationally-oriented American calculus texts than the more theoretical approach that you describe.2011-08-17
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    @3Sphere: I edited my question to answer your comment, I'm sorry I can't do more. I'm still a student and I don't have that much knowledge of published books.2011-08-17
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    @Andy Thanks for the elaboration2011-08-17
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    Thanks for your answer. Concerning "The two other courses are in the second year and are way more theoretical than these two; let me know if you think they are relevant", I would say that for me it is not necessary, but for others it might be useful.2011-08-18