## Monday, October 21, 2019

### Differentiation and Integration

Recently there was an excellent xkcd about differentiation and integration, see here.

This brings up thoughts on diff and int:

1) For some students Integration is when math gets hard.

Diff (at least on the level of Calc I) is rote memorization. A computer program can EASILY do diff

Integration by humans requires more guesswork and thought, Computers can now do it very well but I think that it was  harder to get to work.

Someone who has worked on programs for both, please comment.

2) When I took Honors Calculus back in 1976 (from Jeff Cheeger at SUNY Stonybrook) he made a comment which really puzzled the class, and myself, but later I understood it:

Integration is easier than Differentiation

The class thought this was very odd since the problem of, GIVEN a function, find its diff was easier than GIVEN a function, find its int.  And of course I am talking about the kinds of functions one is
given in Calc I and Calc II, so this is not meant to be a formal statement.

What he meant was that integration  has better mathematical properties than differentiation.  For example, differentiating the function f(x)=abs(x) (absolute value of x)  is problematic at 0, where it has no problem with integration anywhere (alas, if only our society was as relaxed about integration as f(x)=abs(x) is).

So I would say that the class and Dr. Cheeger were both right (someone else might say they were both wrong) we were just looking at different notions of easy and hard.

Are there other cases in math where `easy' and `hard' can mean very different things?

1. In a paper written by Ker-I Ko (survey I believe) it is stated that computing the definite integral of a polynomial time computable real function is #P-hard so formally integrating is very hard even for computers isn't it?

2. Well, symbolic differentiation is easier than symbolic integration.

3. Liouville solved this already. It's the basis of MACSYM and Maple and Mathematica ...

https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)

It always surprises people that "strange functions" like sqrt(tan(x)) actually have an anti-derivative in closed form.

It also matters which field you're working over. In the complex domain, you need to make sure you're sitting on the "correct" Riemannian manifold.

4. In calculus class, differentiation seems to be computationally easier than integration. To Jeff Cheeger, integration is "easier" than differentiation because the set of integrable functions strictly contains the set of differentiable functions. After all, there are functions for which integration is possible and differentiation is impossible (and not the other way around), so integration must be easier, right?

In the same way, P is computationally easier than EXP, but EXP strictly contains P. There exist problems for which solution in exptime is possible but solution in polytime is impossible (and not the other way around), so solution in exptime is "easier."