### More Tensor Index Identity Proofs: EM II Notes 2014_08_18

Summary:  Having worked through the examples that looked the most difficult, today's notes contain examples that are pick-up work from the easy problems.  These are simple-ish tensor index identities, including the divergence of the position vector, the cross product of the position vector, the Laplacian of one over the displacement squared, and the curl of a gradient.

$\nabla \cdot \vec{r} = 3$
$= \dfrac{\partial}{\partial x_i} r_i$
Keep in mind that $r_1 = x$, $r_2 = y$, and $r_3 = z$.  Using the rules of partial differentiation, when the partial operates on the variable it is with respect to it will return 1, and when it operates on any other variable, it will return 0.  The results sum to 3.

$\vec{\nabla} \times \vec{r} = 0$
$=\epsilon_{ijk} \partial_j r_k$
$= 0$

For the $\epsilon{ijk}$ to evaluate to a non-zero result, $j$ and $k$ have to not be equal.  However, as discussed above, if $J \ne k$, then the partial derivative evaluates to zero.  Consequently, the entire expression evaluates to zero.

$\nabla^2 \dfrac{1}{r} = 0$
The trick here is to do the derivatives one at a time, keeping things in index notation and look for things to cancel out.  There's also one other identity we'll need $r^2 = x_i x_i$, where the $x_i$ are the Cartesian components of the coordinate system.

So,

$\nabla^2 \dfrac{1}{r} = \partial_i \left(- \dfrac{x_i}{r^3}\right) = -\dfrac{3}{r^3} + \dfrac{3x_i x_i}{r^5}$,

but, $x_i x_i = r^2$, so the r.h.s. above is 0.

For multipole work where you're taking partial derivatives in multiple dimensions, this comes in handy for expressions like:

$\delta_{ij}\partial_i \partial_j \partial_k \dfrac{1}{r}$,

because the terms can be rearranged to show that any such expression is 0.  For way more detail, check out the material near equation 6.26 in https://drive.google.com/file/d/0B30APQ2sxrAYcHl2R3pCSG1HQXM/edit?usp=sharing

$\vec{\nabla} \times \vec{\nabla}f = 0$

$= \epsilon_{ijk} \partial_j \partial_k f$

The trick here is to think about what terms will survive and what the Levi-Civita symbol will do to them negative sign-wise.  Only pairs of derivatives where $j \ne k$ will survive the Levi-Civita.  There will be two of each of these terms, but they will be of opposite signs and will cancel, for example,

$\epsilon_{i23}\partial_2 \partial_3 = -\epsilon_{i32}\partial_3 \partial_2$.

Hence, all terms will cancel and we have a zero result, and a handy identity moving forward:

$\epsilon_{ijk}\partial_j\partial_k = 0$

Picture of the Day

### 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…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…

### 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…