Saturday, January 26, 2013

The derivative of products

An elementary proof 

Knowing some important formulas by heart can be very useful, but if one knows how to derive them, it is no longer necessary to remember the formula. From reading math textbooks (many  or most of them?), one can gain the false impression that the process of deriving a formula follows the same sequence as the proof.

Here is an elementary proof that nevertheless involves some not so obvious steps. 

Suppose that


then the formula for the derivative is 


but how do we prove this? Although the proof is straightforward, it is perhaps difficult to remember all the tricks that are required and when to apply them. Here is a standard proof. First, apply the definition of derivative to the product of the two functions:



The next step is the crucial operation, at once trivial and far from obvious. We are going to both subtract and add u(x)v(x+h) and rewrite the ratio as


Now, who would think of adding two terms that sum to 0 into such an expression? This is an idea that doesn't make much sense at this point. Indeed, one needs to look a few steps ahead and see what it is going to be needed for. What follows are just some simple factorizations of terms.


Break out some terms to get 


then take limits and replace derivatives, and we are done:




Here, the simple formula seems much easier to memorize than all the steps of the proof. (In fact, you may impress your friends far more if you memorize Hugo Ball's poem Karawane than if you learn to recite the steps of this proof.)

It is highly misleading when formulas such as the above are just plainly stated and then concisely proven. This is most likely not how the formulas were originally discovered. Rather, one would observe a few instances of derivatives of multiplied functions and conjecture a formula. Then, starting from the formula as well as the definition of derivative, one would work backwards and find all the arithmetic manipulations that make the proof work.

Instead of learning a fixed set of steps that are used in particular proofs, one would probably learn a bag of tricks that can be applied in various situations. Then, out of this bag one can grab various operations that can be tried out, until something is found that leads the proof in a promising direction.


Tuesday, January 15, 2013

Against gadgetry


Purportedly intelligent functionality increasingly finds its way into consumer electronics of all kinds. Consider video cameras as an example. Instead of manual brightness and focus controls, these can be handled automatically by pointing the camera at the right target. Fine, except that this makes it more difficult to gain control over the footage if the automation cannot be overridden.

In theory, it would be possible to have a function on your camera that finds out the name of a person whose face you point it at. Similar risks and annoyances are likely to crop up in all places where too much electronic connectivity is built into products.

Someone said that a good music instrument is one that allows you to play badly. It doesn't correct your mistakes, so you have to practice. If you have practiced and try playing a gadget that corrects your mistakes, it will only stand in your way.

There are many reasons to keep things simple and stupid. Open modular systems is one interesting trend offering the perfect antidote to these over-designed digital marvels.


Friday, January 11, 2013

Antropocene

An excellent resource for understanding the current understanding of climate change is the recent series of blog posts by John Baez. Whereas journalists oversimplify matters, diving straight into the research papers would be overwhelming. Baez is the ideal guide if you know some elementary mathematics (just the basics of ordinary differential equations will do). He begins by explaining concepts such as albedo and the energy balance due to incoming and reflected sunlight. Then some positive and negative feedback mechanisms are introduced, and there is a discussion of glacial cycles, bistable models and stochastic resonance. All of it is very accessibly explained; it is definitely worthwhile to take the time and digest this material.

It begins here with some slides and continues as a series of blog posts.

Another useful source of information is skeptical science, especially if one ever needs to debunk the myths that people pass on without checking the facts.