### Levi-Civita Product to Kronecker Delta Difference of Products: EMII Notes 2014_08_08

Summary of what's gone on before.  In the previous set of notes from the 6th, (there were no notes on the 7th), it was pointed out that the 'convenient' comment on page 11 of the notes was to cryptic.  Today's entire half hour was spent figuring out the following derivation that sprang from the convenient comment.  We want to derive:

$\epsilon_{ijk}\epsilon_{lmk} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}$

Here's what to do.  First remember sum notation gives you

$\epsilon_{ijk}\epsilon_{lmk} = \epsilon_{ij1}\epsilon_{lm1} + \epsilon_{ij2}\epsilon_{lm2} +\epsilon_{ij3}\epsilon_{lm3}$

Here's the first use of the big trick for the day.  Because of the properties of the Levi-Civita symbol, $\epsilon_{ijk}$, on the indices 2 and 3 will make the first term non-zero, while only the pairs 1,3 and 1,2 will make the other two terms non-zero.  Once any of these combinations is chosen however, the other two terms will vanish.  Given that, let's get to work on the first term, ignoring the other two that evaluate to zero.

$\epsilon_{ij1}\epsilon_{lm1}$

For this expression, the only index choices that will evaluate as non-zero are shown in the table below:

Index Choices and Results

 i j l m Result 2 3 3 2 -1 2 3 2 3 +1 3 2 2 3 -1 3 2 3 2 +1

At this point, we have to realize tht we have two indices that reutrn either one or negative one.  This should bring the Kronecker delta to mind.  Note that when $i=l$ and $j=m$ we get $+1$, while when $i=m$ and $j=l$ we get $-1$.   We can then right these combinations as

$\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$

So,

$\epsilon_{ijk}\epsilon_{lmk} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}$

and we're done.

Handy reference that I found after doing the work:

The dudes:

### 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…