Application of Harmonic function

Posted by tpc at May 11th, 2010

Cool article about Pixar and some applications of harmonic function. Would have been useful last year when I taught the maximum principle but didn’t have much else to say about it.


Moving Remy in Harmony: Pixar’s Use of Harmonic Functions
 by David Austin.
via: JohnDCook’s twitter.

Posted in Applications, Calculus/Analysis, Complex Numbers, Teaching, Technology| No Comments | 

Breaking up is infinitely hard

Posted by tpc at January 24th, 2009

I was reading this light-hearted article by jeremy au yong in the local papers. It’s mainly about how the economy is bad and he was retrenching his girlfriend. As compensation he offered a gift.

The value of this gift will be determined by using the formula $15 x years of service + cube root of pi, differentiated over the limit of zero to infinity.

Something is wrong right? Does he mean integrated rather than differentiated, since nobody differentiates over a range. As the only variable is y for years of service, does he mean
\displaystyle \int_0^\infty (15 y + \sqrt[3]\pi ) dy = \infty.

Well, how does one pay off this infinite debt? Once again, mathematics to the rescue. He can pay his girlfriend 1 cent today, 1/2 cent tomorrow and 1/3 cent the day after …
\displaystyle \sum_{n=1}^\infty \frac{1}{n} = \infty.

Posted in Calculus/Analysis, Fun Stuff| No Comments | 

Separating the variables

Posted by tpc at January 13th, 2009

We should all be familiar with the method of separation of variables for first order ordinary differential equations. Here’s a neat example from John Starrett’s article in Amer. Math. Monthly.

\displaystyle \frac{dy}{dx} = \frac{ y^3 + x^2y -x -y}{ x^3+ xy^2 -x+y}
(more…)

Posted in Calculus/Analysis, Problems| No Comments | 

The Arithmetic vs Geometric Mean Trick

Posted by tpc at February 22nd, 2007

One classical trick is the following:
Since
(x -y)^2 \ge 0, we have
x^2 - 2 x y + y^2 \ge 0 \implies x^2 + 2 x y + y^2 \ge 4xy
Taking root we obtain (x+y)/2 \ge \sqrt{xy}

A variation of this trick can be used to show that
| a \sin x + b \cos x | \le \sqrt{a^2 +b^2}.
We start with (a\cos x - b \sin x)^2 \ge 0. Expanding the expression we get
a^2 - a^2 \sin^2 x - 2 a b \sin x \cos x + b^2 - b^2 \cos^2 x \ge 0.
A simple rearrangement completes the proof.

Posted in Calculus/Analysis| 1 Comment | 

Silly Riddle

Posted by tpc at February 22nd, 2007

What’s the difference between jumping down from the 2nd floor versus the 20th floor?

Ans:
Splat! Aaaaaaahhhhh! (2nd floor)
Aaaaaaaahhhhh! Splat! (20th floor)

What if you jump from the 10th floor? It all depends on how long you remain in the air. A little calculus (plus some physics) does the trick.

Let v be your velocity and u be the initial velocity which is assumed to be zero. That is you gently let yourself off the edge instead of making running jump. Of course a will be acceleration 10 m/s^2. Let’s not be fussy about precision.

Let t_s, i.e. time_splat, be the time you hit the ground and d the distance covered. We have
d = \int_0^{t_s} v dt = \int_0^{t_s} u +at \: dt = ut_s + \frac{1}{2} a t_s^2

Now let’s assume average height per floor is about 3 metres, so d = 30 gives
t_s = \sqrt{6} .
About 2.5 seconds, I guess enough for Aahh!Splat!
There you have it, who says calculus is useless?

Posted in Calculus/Analysis| No Comments | 

Integration problem from pAt84

Posted by tpc at December 27th, 2005

My latexrender (installed by my wife with much pain) has gone to disuse. So I’m putting up this problem posted by pAt84.

Version 1
\mbox{\Large \mu = \int_0^1 \frac{ \big( \sum_{j=0}^n a_j t^j \big) \big( \sum_{i=1}^n i b_i t^i \big) } { \sqrt{\big( \sum_{j=1}^n j a_j t^j \big) ^2 + \big( \sum_{i=1}^n i b_i t^i \big)^2 }} dt}

Version 2
\mbox{\Large \mu = \int \frac{p(x) q\prime (x)}{ \sqrt{ p\prime (x)^2 + q \prime (x)^2} } dx}

Posted in Calculus/Analysis, Problems| 1 Comment | 

Stoke’s Theorem and Vector Analysis

Posted by tpc at April 10th, 2005

When I first learnt this 9 years ago, I hated it. It was a “here’s the formula, use it” kind of course. Second time around, I learn the subject in the context of differential forms and boy what a world of difference.

\int_{\partial\Omega} \omega = \int_{\Omega} d\omega

and

\Lambda^0 \; \stackrel{ grad } {\longrightarrow} \; \Lambda^1 \; \stackrel{ curl } {\longrightarrow} \; \Lambda^2 \; \stackrel{ div } {\longrightarrow} \; \Lambda^3

is indeed a most beautiful theory.

The funny thing was that I had to teach two tutorial sessions on the same subject while I was relearning the material, and so I had a very clear idea of what was happening. But how much did that benefit my students, I would never know.

Posted in Calculus/Analysis| 6 Comments | 

Elementary Proofs

Posted by tpc at February 8th, 2005

No one shall expel us from the paradise that Calculus has created.
- tpc

The fact that an analytic solution to number theoretical problems exist is bewildering. And often, the analytic proof is much simpler than the elementary proof. The Prime Number Theorem is the most famous example. Another is the evaluation of \zeta(2) and \zeta(3) in terms of double integrals.

The following is strictly not a number theory problem:
A rectangle is divided into smaller rectangles. Each of the smaller rectangles has the property that at least one of the sides has integer length. Show that the large rectangle has the same property.

I first encountered the problem at kuro5hin via Chapterzero, but apparently it dates back to de Bruijin in the sixties. I couldn’t solve it but as luck would have it, saw a solution in Paul Zeitz’s The Art and Craft of Problem Solving, which provided a reference to an article in American Math Monthly (vol 94) by Stan Wagon which gave 14 proofs, including a one line calculus proof.

Posted in Calculus/Analysis, Number Theory, Problems| 2 Comments | 

Irresistible Integrals

Posted by tpc at January 24th, 2005

A book by George Boros and Victor Moll. The authors bill the book as a guide to the evaluation of integrals, but it strucked me as a nice guide to old-fashioned (nineteen century) analysis, in the spirit of Gauss and Euler, if I may add. There are many gems in the book including three proofs of
\sum_{n=1}^\infty \; \frac{1}{n {2n \choose n} } = \frac{\pi}{3\sqrt{3}}
and also this which is not proved
\sum_{n=1}^\infty \; \frac{(-1)^{n-1}}{n {2n \choose n} } = \frac{2 \tau}{\sqrt{5}}
where \tau = \log((1+\sqrt{5})/2).

Posted in Books, Calculus/Analysis| 1 Comment | 

Irritating Integral

Posted by tpc at November 28th, 2004

It took me almost a week to prove \int_0^{\pi} \log (1+4\cos^2(u)) du = 2\pi \log{\phi} where \phi = (\sqrt{5}+1)/2.

A time consuming but fruitful exercise. I had tried all the integration tricks I know, resorted to computer packages Maple and Mathematica - Integrator . In the end, I used contour integration and it took another two days to find the correct contour. I may not publish my solution here, in case I meet some smug kid who badly need to be tormented.

I found the problem here, the given solution is a heuristic argument.

Posted in Calculus/Analysis, Problems| No Comments |