0

In my Analysis class we keep using the symbol $D$ to stand for differentiation in our analysis course, but what is $D$ itself, really?

  • nLab seems to say its a functor, but for some reason requires smoothness (as oposed to e.g. $C^1$ or even just differentiability).
  • Dieudonné, in his Treatise on Analysis, Vol. 4, p. 127, talks about using Lie Groups to generalize "the operators of differentiation".
  • Bourbaki, in their Lie Groups and Lie Algebras, Ch. 3 §17, have this definition:

screenshot

  • This paper uses "the differentiation operator" in the context of "weighted spaces of holomorphic functions".
  • This other paper says "the differentiation operator $D$ is defined by $Df=f'$, [...]"
  • There are some related questions on here too:
  • Wikipedia talks about differential operators generalizing the differentiation operator. It also has this article about generalizing the derivative.
  • Of course, many authors just introduce $D$ as a notation and then move on without mentioning it again.

I literally just want to know what $D$ is so that I can continue with my analysis lectures without feeling like I'm already using things without knowing what they are. I tried asking something similar before (see here), but I suspect the question was unclear.

In the very possible case that there's different approaches to what $D$ might be (i.e. different, incompatible generalizations), some clarity would still be much appreciated. Thank you!

steve
  • 302
  • 4
    It is whatever the author of the book decides it means. If I find a “D” in some algebra book, it could denote something related to (formal) differentiation, but it could just be some module or submodule. – egreg Apr 26 '20 at 12:56
  • 3
    Letters are not always used in the same way! There are many "usual" ways in which the letter $D$ is used in analysis, many contradicting each other! Thats not a problem, as long as you are making sure that people reading know what you mean by setting up notation carefully. – s.harp Apr 26 '20 at 12:57
  • @egreg Well, yeah, of course, but since so many different authors (in different fields, even) chose to generalize this very same letter in one way or another, the question really is which generalization is compatible with its usage in an analysis textbook. (That it's just a letter by itself and is often used to denote other arbitrary objects in mathematics is true, of course, but I don't think it stands in the way of the question.) – steve Apr 26 '20 at 13:01
  • 4
    The letters $D$ and $d$ stand for "differential" or "derivation" in these contexts and it is really natural and established to use them when you have something you call a differential. In analysis and geometry I like to use the letter $D$ for the derivative of a map, ie if $f:M\to N$ then $Df: TM\to TN$ is its derivative. The letter $d$ is usually used for the exterior derivative in this context. But its really entirely arbitrary and people do what they want. There is no canonical meaning to "D". – s.harp Apr 26 '20 at 13:05
  • @s.harp So e.g. assuming your definition of $D$ (which, while perhaps not canonical, does seem to be pretty well-liked), you'd say that $D$ is a function between function spaces? Or a functor? Or does the literature really just use it as a notation with little further motivation behind it? – steve Apr 26 '20 at 13:08
  • 1
    That concrete definition satisfies the property of a functor. Its also a function between function spaces, but I don't think thats a very useful way of thinking about it. – s.harp Apr 26 '20 at 13:11
  • @s.harp Assuming that'd be the same functor as the one from nLab, why is smoothness required? Or, if it's also a function between function spaces, what function spaces would those be? Is there a general yet compatible way to define it? – steve Apr 26 '20 at 13:16
  • 1
    Smoothness is not required for the formulation. It could be a functor between the category of $C^k$ manifolds with $C^k$ maps as morphisms and the category of $C^{k}$ manifolds with $C^{k-1}$ bundle maps on their tangent bundles. – s.harp Apr 26 '20 at 14:44
  • 2
    @BlondCafé You haven't given us the context of what kind of analysis class you're in, but the simplest possible interpretation is that $D$ is a (linear) function from the vector space of differentiable functions on $\mathbb R$ to the vector space of functions on $\mathbb R$ which admit an antiderivative. In any case, the conclusion you should draw from all these comments along the lines of "impossible to say without more info" is that you should ask your professor. – Kevin Carlson Apr 26 '20 at 19:21
  • @KevinCarlson That's a very fair point, I'm sorry I didn't provide context. It's a basic (Real) Analysis II course, i.e. inching towards differentiation of functions between Banach spaces, but right now still only dealing with $\mathbb{R}^n$. The question as I meant it is out of scope for the course though; it's just that I sometimes feel the need to know what lies behind the notations and conventions (even if what's "behind" might be more recent). To me, the most general form of a concept is its simplest and most satisfying, even if its rarer, more advanced, or just verbose. [continued] – steve Apr 26 '20 at 19:37
  • [continuation] And so the idea of differentiation being a functor is very appealing to me (esp. if no further requirements like smoothness is required, though I'm having difficulty finding much (any, in fact) literature with this approach), as is the idea of it having to do something with Lie theory, or other generalizations. It's like e.g. generalizing the notion of a limit with nets or with filters. A good recommendation for some work that focuses on this (instead of possibly just using it as a short example or so) would also, of course, be much appreciated. – steve Apr 26 '20 at 19:38
  • @BlondCafé Well, if you want to ask "what are some generalizations of differentiation", that doesn't sound quite like the question that was given. But a reference for the kinds of things you're saying is essentially just "any book on differential topology." It's not clear that there's anything very specific to say. – Kevin Carlson Apr 26 '20 at 21:13
  • @KevinCarlson Okay, that's a very fair point yet again. I must admit my confusion / desperation got the best of me. Also, I'll ask my Professor to see what he has to say. May I ask you if you think I should delete this question and ask a new one if necessary, or just edit this one, or just leave it as is? – steve Apr 26 '20 at 21:31
  • @BlondCafé If you see a way to make essentially this question a bit more specific, then you should edit this one. – Kevin Carlson Apr 26 '20 at 21:35

0 Answers0