Revision of To understand an object, consider treating it as one of a family of objects and analysing the family from Sun, 26/04/2009 - 18:24

Quick description

Sometimes an individual object may be difficult to understand in isolation, but easier to understand if you treat it as a member of a family of objects of a similar type. This is a huge theme in mathematics: in this article we give a brief description of it, and links to articles about its manifestations in different areas.

This article is incomplete. More suggestions wanted for subarticles. Expert attention needed. It would be nice if the subarticles could be written too.

Example 1

The integral $\int_0^{2\pi}e^{\cos(\theta)}\cos(\sin(\theta))d\theta$ is not very easy to evaluate directly. However, it can be evaluated as follows. First, you look at the more general integral $\int_0^{2\pi}e^{\phi\cos(\theta)}\cos(\phi\sin(\theta))d\theta$ , which we shall refer to as $f(\phi)$ . Next, you differentiate $f(\phi)$ with respect to $\phi$ . The derivative turns out to be $0$ . Since $f(0)=2\pi$ , it follows that the original integral, which equals $f(1)$ , is also $2\pi$ . The details of this calculation can be found in Example 5 of the Wikipedia article on differentiation under the integral sign.

General discussion

The relevance of the above example to this article is that we started with just one integral, but in order to understand it we treated it as a member of a 1-parameter family of integrals $0\leq t\leq 1\}$ . This family has a certain structure to it, which allows us to use a tool, namely differentiation, that was unavailable to us before. And that led to a straightforward solution to the original problem.

It is important to understand the difference between this and mere generalization. Here, we are generalizing in a very particular way, by embedding an object into a structured family of objects. This allows us to exploit relationships between members of the family.

Possible subarticles

Clearly there should be an article on the use of moduli spaces. Also, there should be one on families of zeta functions. If anyone wants to create stubs or rudimentary articles for these, that would be great. And of course there must be lots of other potential articles that would fit into this general theme.