What is the importance of Calculus in today's Mathematics?

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.

[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.

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.