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I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is defined. This lets you generalize these ideas to spaces well beyond the simple $\mathbb{R}^n$ of early calculus. My question is, what sort of structure do you need to define a derivative? I'm familiar with the idea of a differential ring, but I'm looking for something defined on functions. Specifically, given a base space $X$ and a target space $Y$, I'd like some operator $D$ that takes certain functions $f:X\rightarrow Y$ and yields a derivative function $Df=f':X\to Y$. This should agree with the usual definition of the derivative on Euclidean space, so it should be linear, obey the chain rule, etc.

What structure do you need before this $D$ operator can exist? Will we need so much structure that we're forced to make our spaces Euclidean? And if so, could we loosen the requirements for our "derivative" to get some more generality? Any help and input would be appreciated!

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    $\begingroup$ In a certain perspective, the elements of any ring are functions. $\endgroup$ – curious Nov 9 '14 at 3:26
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    $\begingroup$ Also, smooth manifolds are an obvious choice, but the derivative of a smooth map $f:M\to N$ is a map $Df:TM\to TN$, and it sounds like you would want the derivative to again be a map $M\to N$. Do I understand your goal correctly? $\endgroup$ – curious Nov 9 '14 at 3:28
  • $\begingroup$ As to the first comment, do you mean a ring of functions or a Cayley's-theorem-style representation? $\endgroup$ – Zach Effman Nov 9 '14 at 3:31
  • $\begingroup$ And yes, I would prefer something a bit more general than smooth manifolds. I'd accept some sort of tangent space construction as long as it still gave $T_p(\mathbb{R}^n)\cong \mathbb{R}^n$, but it would be nice to avoid it if possible. $\endgroup$ – Zach Effman Nov 9 '14 at 3:32
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    $\begingroup$ Using the Frechet derivative, you can at least generalize the concept to arbitrary Banach spaces. $\endgroup$ – PhoemueX Nov 9 '14 at 5:49
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There are lots of ways to generalize the derivative, not all of which are compatible. This is a common feature of generalizations: for example, there are also lots of ways to generalize numbers, or infinity, or exponentiation, not all of which are compatible. Here are some generalizations of derivatives you can write down:

  1. The Fréchet derivative of a map of Banach spaces. This is used in the theory of Banach manifolds.

  2. A derivation on an algebra. This abstracts linearity and the Leibniz rule and can be used to define tangent vectors and vector fields in a very general setting; in particular it can be used to describe what it means for a Lie algebra to act on an algebra.

  3. Mapping out of infinitesimal objects in suitable categories gives many notions of the derivative of a map between objects. For example, in the category of schemes over a field $k$ the "walking tangent vector" $\text{Spec } k[\varepsilon]/\varepsilon^2$ has the property that mapping out of it corresponds to taking the derivative of a map of schemes in the sense that it describes the induced map on all Zariski tangent spaces.

  4. There is a combinatorial notion of the derivative of a combinatorial species that, upon taking exponential generating functions, recovers the derivative of a power series. It is a combinatorially meaningful operation and requires no calculus to describe.

Incidentally, integration also has generalizations like this. For example, there is a completely algebraic approach to integration involving writing down a linear functional on an algebra satisfying some conditions. It has the very desirable property of naturally including "noncommutative probability" as a special case, where there is no underlying measure space but there is still a useful notion of a (noncommutative) algebra of random variables and of expectation values of these. See this blog post for some details. This can be used to explain some aspects of quantum mechanics, and it also leads to free probability.

This is even true of limits. Some notions of convergence, such as the notion of almost-everywhere convergence of a sequence of functions on a measure space, aren't defined by a topology! See convergence space for some details.

I guess the point I'm trying to make here is that you can generalize things in all sorts of ways, depending on what you're trying to do. Don't feel burdened by the first formalism you see for doing something.

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