### tan(x)=x The Overlooked Approximation and Other Notes

My quantum professor always knows exactly the right approximations to make, (I suppose it helps to be the person who wrote the homework).  Regarding the recent homework questions involving boundary conditions in shallow square potential wells which are just oozing with tangents of ratios of wave numbers, he pointed out a trig approximation that's often overlooked, (at least by students like myself).  At small angles (picture 1)

the tangent of x is just x.

Yup, small angles aren't just for sine anymore!  The tangent of a small angle is also equal to it's argument.  I had to go through the additional little thought process of (picture 2)

Here's a graph of tan and sin overlayed with x so you can see what's going on. (picture 3)

The Levi-Civita Symbol
Another mathematical tool that has come up repeatedly in the last few weeks is the Levi-Civita symbol.  If you've studied the cross product or the vector curl you've seen it.  Wikipedia defines the symbol like this (picture 4)

They go on to say that it's +1 if the indices are an even permutation of (1,2,3), and -1 if the indices are an odd permutation of (1,2,3).  My rule for remembering this is that if you can read the indices in order from left to right, (wrapping when you hit a close paren), then the results is +1.  If you can read the indices in order from right to left, (wrapping when you hit an open paren), then the result is -1.  Positive numbers on the number line go from left to right and negative numbers go from right to left... Voila.

This symbol turns up over and over again and is very handy to memorize.  When you want to put things succinctly, and impress your friends, you can write down the components of a cross product, or a curl, or any number of electromagnetics equations using the Levi-Civita symbol.  For example, Wikipedia defines the vector curl  as (picture 5)

References:
1.  http://en.wikipedia.org/wiki/Levi_civita_symbol

2.  xkcd style graphs:
http://mathematica.stackexchange.com/questions/11350/xkcd-style-graphs

Picture of the Day:
Today's pictue of the day is taken from Wikipedia and is of Tullio Levi-Civita.  He's one of the few 19th and 20th century scientists I've researched lately who was actually smiling!  He's cool! (picture 6)

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…