Archive for October, 2013

Sublime Text Tip: Indicating Current and Edited Tabs

Friday, 25 October 2013

To colour the current tab green, and any tab of an edited file red, add the following to your theme file.


Hyperbolic Mnemonic

Friday, 11 October 2013

If you have problems remembering which is which with sinh and cosh, or their graphs, or which way around their definitions are:

\sinh x = \dfrac{\mathrm{e}^x - \mathrm{e}^{-x}}{2}

\cosh x = \dfrac{\mathrm{e}^x + \mathrm{e}^{-x}}{2}

then it might help if you notice that the graph of sinh is somewhat S-shaped (though backwards), and cosh is C-shaped (though open upwards).

[Well, unless you’re Russian, where the Cyrillic letter Es (‘С’) corresponds to our ‘S’; that might throw a graphemic spanner in the works.]

Derivation Summary

Friday, 11 October 2013

Here is a summary of rules of derivation (sometimes referred to as differentiation).

I wrote this post as the notation used on other sites is often ambiguous (particularly with juxtapositions that could denote either a product or a function application).

I previously wrote some brief notes on some derivations of the derivation formulae.


Surface Areas and Volumes of Hyperspheres

Thursday, 3 October 2013

I noticed that the formulae for the sizes of the surface (or rather, boundary) and volume (interior) of circles, spheres and hyperspheres seem to have discontinuities (or ‘jumps’) in the index of \pi.

However, this is only a result of simplifying a uniform formula for particular cases.


On Fibonacci-style Sequences continued

Tuesday, 1 October 2013

In my previous post on Fibonacci-style sequences, I gave the formulae for the closed form

a_n = p \cdot r^n + q \cdot s^n

given the recurrence values a_0, a_1, b, c where

a_{n+2} = b \cdot a_{n} + c \cdot a_{n+1}

but I did not show that the formula is valid. I do that here.