### Index Notation Partials and Rotations, a Cool Gradient Trick: EMII Notes 2014_08_09 Part I

Summary of what's gone on before.  Finally got the Levi-Civita to Kronecker delta identity down yesterday, 2014/08/08.  Today we're making more use of the index notation.  There are however, some notational stumbling points

$r^2 = x^2 + y^2 + z^2$

$\vec{r} = \left(x, y, z\right)$

Now if we want to take the derivative of both sides of the magnitude equation above, first remember we can write $r^2$ as

$r^2 = x_jx_j$

Now, finally taking the derivative of both sides o the above we get

$2r \partial_i r = 2x_j\partial_i x_j$

remembering the partial differentiation rules and the kronecker delta we can write the above down as

$2r \partial_i r = 2x_j \delta_{ij} = 2x_i$

which finally gives:

$\partial_i r = \dfrac{x_i}{r}$

The Kronecker trick above is crucial. Also remember, not one of the r's is a vector, they're all the magnitude of the vector.

\section{Rotations}

Any rigid rotation of a vector can be defined as:

$\begin{pmatrix} x'\\ y'\\ z' \end{pmatrix} = M\begin{pmatrix} x'\\ y'\\ z' \end{pmatrix}$

Here are a few of the important parts.  The Matrix $M$ is orthogonal and, $M^TM = 1$

The transpose actually defines orthogonality.  If such a matrix is of dimension n, then it is n $)\left(n\right)$ matrix.

The footnote has all the cool kid stuff about rotation matrices and how to name them:

First of all, if the determinant of a matrix is +1, then it's a special orthogonal matrix, $SO\left(n\right)$.  The other sort, the sort with a determinant of -1 are actually rotations with a reflection of the coordinates.

There are also some identities in the footnote that will come in handy

$det\left(AB\right) = \left(det A\right)\left(B\right)$

and

$det\left(A^T\right) = det\left(A\right)$

Using the above two identities, we can see that $\left(det M\right)^2 = 1$

Memorize these!!!

In index notation, the rotation above can be written as
$x_i^{\prime} = M_{ij}x_j$

The orthogonality condition becomes

$M_{ki}M_{kj} = \delta_{ij}$

### 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…