## Potential energy in classical mechanics

Posted Jun19-13 at 04:01 AM by CompuChip

Actually the idea of potential energy is particularly useful for so-called conservative forces. Crudely put, these are the forces whose work done only depends on your position, not how you got there. For example, gravity is a conservative force: whether you move straight up and down, or in a circle, or in a spiral path to Mars and back - when you end up at your initial position you will have exactly the same amount energy. In contract, a force like drag or friction is not conservative: the longer...
Posted in Uncategorized

## The generalised three-door problem

Posted May16-11 at 12:22 PM by CompuChip

Today I was having a discussion with a colleague about the three-door problem. I'm sure you've all heard of it in some form or another. Most commonly it is given as a game show concept, where a contestant has the choice between three doors. One conceals some prize (a car, or a large sum of money) and the other two don't (they contain a goat or nothing at all). After choosing one of the doors, the game show host opens one of the remaining two doors and shows that the big prize is not behind it. Then...
Posted in Uncategorized

## Taylor series: from linear to higher order approximation

Posted Feb27-10 at 06:25 AM by CompuChip
Updated May30-10 at 04:52 AM by CompuChip (Fix some small typo's/mistakes)

So the idea of a linear approximation, is to describe a function f in the neighborhood of some point a by a linear function. In terms of the graph of f, we want to find a straight line, such that in the vicinity of a, we can approximate the function by the line. If you think about this for a moment, you will realize that we are talking about the tangent line here. Thus you quickly arrive at the equation
y = f(a) + f'(a) (x - a).
Staring at this, you may realize that f'(a)(x - a) is...
Posted in Uncategorized

## Proof by induction

Posted Sep27-09 at 09:31 AM by CompuChip

Generally, when you have some statement like "for all n, X is true", a proof by induction consists of two steps. First, you have to show that for some simple case (usually n = 0 or 1, depending on the question), X is true. Then you assume that X is true for all integers n up to some given value n0, and you prove that under that assumption, X is also true for n0 + 1.

The reasoning is then as follows: you have checked by hand that it is true for n = 1. You have proven that...
Posted in Uncategorized

## Installing LaTeX on Windows

Posted Jul22-09 at 04:39 PM by CompuChip
Updated Jan11-12 at 08:07 AM by CompuChip (Added Linux distro)

Using LaTeX is a bit like programming. You need a "compiler" (TeX distribution) to convert the documents you type to a final format, such as PDF. For Windows, MikTeX is the most common and easiest to install, in my experience. For *NIX (e.g. Ubuntu) texlive is straightforward to install using apt.
The documents themselves can be typed in whatever program you like, you can do it in Notepad for all anyone cares. However, there are several programs available which make editing TeX...
Posted in Uncategorized