### Two More Tensor Identities, and then Special Relativity Next! EM II Notes 2014_08_22

Summary:  Still More Tensor Identities
Three more identities with gradients, divergences, Laplacians, and cross products.  Later today, the fun stuff, special relativity begins!

$\vec{A} \cdot \left(\vec{B} \times \vec{C}\right) = \vec{B} \cdot \left(\vec{C} \times \vec{A}\right)$

$= A_i \epsilon_{ijk} B_j C_k$

As long as we only cycle the indices in the Levi-Civita symbol, we won't cause a sign change, so the above is also equal to

$= A_k \epsilon_{ijk} B_i C_j$

Which we can commute to get

$= B_i \epsilon_{ijk} C_j A_k = \vec{B} \cdot \left(\vec{C} \times \vec{A}\right)$

Done!

$\vec{\nabla} \cdot \left(\vec{\nabla} \times \vec{A} \right) = 0$

$=\partial_i \epsilon_{ijk} \partial_j A_k$
$= 0$

If $i$ and $j$ are equal, then the Levi-Civita evaluates to zero.  If they are not equal, then swapping the two indices produces the same mixed partial derivative result, but with a negative sign inserted by swapping indices in the Levi-Civita symbol.  These equal but opposite terms all sum to zero giving the advertised result.

$\vec{\nabla} \times \vec{\nabla} \times \vec{A} = \vec{\nabla}\left(\vec{\nabla} \cdot \vec{A}\right) - \nabla^2 \vec{A}$

$= \epsilon_{lmi} \partial_m \epsilon_{ijk} \partial_j A_k$

Don't let all the dels, nablas, whatever you'd like to call them, throw you off.  At the end of the day, this is just the same as the $\vec{A} \times \vec{B} \times \vec{C}$ example.  Use the two Levi-Civitas to four deltas identity.

$= \epsilon_{lmi} \partial_m \epsilon_{ijk} \partial_j A_k = \epsilon_{ijk} \partial_j \epsilon_{lmk} \partial_l A_m = \left(\delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}\right)\partial_j\partial_l A_m$

$=\delta_{il}\delta_{jm}\partial_j\partial_l A_m - \delta_{im}\delta_{jl}\partial_j\partial_l A_m$

$= \partial_i \partial_j A_j - \partial_j \partial_j A_i = \vec{\nabla}\left(\vec{\nabla}\cdot \vec{A}\right) - \nabla^2 \vec{A}$

Picture of the Day:
Hiking trail near Redwood City, CA

### 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…