19 February 2015

Coulomb's law

About a year ago, I discovered a very interesting short video by Richard Feynman responding to the question of what happens when we hold two magnets next to one another. His answer was brilliant.

Recently, I came across a video from the 50s where a physicist is conducting experiments trying to explain electrostatics and, more precisely, Coulomb's force. What is very interesting is that he doesn't merely present facts, but also argues, from various points of view, trying to convince the listener that the force must depend on the distance r between two charges as a linear function of 1/r2 (and linearly on each charge). The argument, especially towards the end, is not dissimilar from that of a mathematician who is trying to explain Potential Theory and Laplace's equation. In the end (skip to 17'22'' if you wish), we realize, with a bit of thinking, that a lot of the things we see in the experiment are merely outcomes of geometry and the fact that we live in a 3 dimensional space (this is an experimental observation that works well!) where distances obey the Pythagorean theorem.

I found out that the physicist is Eric Rogers. The video was intended to be for secondary schools. These days, one can find students of electrical engineering who do not understand electrostatics, students of mathematics who do not understand what potential theory has to do with physics, and educators (they are called pedagogues--and, as I have explained, they form a modern type of plague) who insist that education is independent of the discipline. Not that there aren't universities that teach properly and students who understand a lot, but this species (those who strive to understand and, hence, to explain) is becoming rarer and rarer.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant