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

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).

Answer

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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