Stupid question: why does it appear to us that differentiation is easier than anti-differentiation?

This post is inspired by my rereading a favorite book of mine: Underwood Dudley’s Mathematical Cranks

There was the chapter about the circumference of an ellipse. Now, given it isn’t hard to see that and so going with the portion in the first quadrant: one can derive that the circumference is given by the elliptic integral of the second kind, which is one of those integrals that can NOT be solved in “closed form” by anti-differentiation of elementary functions.

There are lots of integrals like this; e. g. is a very famous example. Here is a good, accessible paper on the subject of non-elementary integrals (by Marchisotto and Zakeri).

So this gets me thinking: why is anti-differentiation so much harder than taking the derivative? Is this because of the functions that we’ve chosen to represent the “elementary anti-derivatives”?

I know; this is not a well formulated question; but it has always bugged me. Oh yes, I am teaching two sections of first semester calculus this upcoming semester.

Differential equations mentioned in National Review

(hat tip: Vox)

The National Review excerpt:

[…]One part insecure hipsterism, one part unwarranted condescension, the two defining characteristics of self-professed nerds are (a) the belief that one can discover all of the secrets of human experience through differential equations and (b) the unlovely tendency to presume themselves to be smarter than everybody else in the world. Prominent examples include […]

(emphasis mine).

Oh noes! I love differential equations! 🙂

Yeah, I am just having fun with the quote; I couldn’t resist mentioning an article in the popular press that mentions differential equations. I am not sure that I’ll teach the chapter on “all the secrets of human experience” in my upcoming differential equations class though.

This is SO me during summer break…

(click for larger)

It is a minor miracle that I publish at all during the “social media” era. 🙂

I am going to celebrate this…

This marks the second summer in a row I got news that a paper of mine has been accepted for publication. Last year, it was the College Mathematics Journal; this year it is the Journal of Knot Theory and its Ramifications.

Sure, that is a big “yawn”, “so what”, or “is that all?” for faculty at Division I research universities. But I teach at a 11-12 hour load institution which also has committee requirements.

And, to be blunt: I got my Ph. D. in 1991 and has a somewhat long slump in publication; I was beginning to wonder if my intellect had atrophied with time.

Ok, it has, in the sense that I don’t pick up material as quickly as I once did. But to counter that, the years of teaching across the curriculum (from business calculus to operations research to numerical analysis to differential equations) and the years of attending talks and attempting to learn new things has given me a bit more perspective. I make fewer “hidden assumptions” now.

So, I am going to celebrate this one…and then get back to work on spin-off ideas.

Parabolas have interesting properties

And this is one of them:

A quickie from Mathematics Magazine

It has been a long time since I’ve posted; I’ve spent time doing various things, including revising a paper that a journal editor wanted revised.

I’ll speak more about that later.

I got the latest Mathematics Magazine in the mail (volume 87, No. 3, June 2014), and the article “Surprises” by Felix Lazebnik is chock full of delightful tidbits, many of which I didn’t know.

Here is a fun one that you can share with your non-mathematically inclined friends (other tidbits there require some mathematics background).

(Surprise 7): A watermelon is 99 percent water (by weight). One ton of watermelons was shipped and during shipment some water evaporated. The watermelons that arrived were made up of 98 percent water (by weight). What was the weight of the shipment when it arrived?

The answer is in the article, but I suggest you give it a go. It will take, at most, a minute or two (if that). This shows the power of basic algebra to discipline our thinking and how our intuition can deceive us.