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### Cosine Laws, Polyhedra, and Legendre Functions

I didn't make it into the lab today what with the holiday and all, but I did have time to read one of my favorite journals, American Mathematical Monthly from the +Mathematical Association of America .  The journal features a very interesting article[1] by Marshall Hampton[3] about cosine identities.  The article got me back to musing about solving for potentials with spherical symmetries and Legendre polynomials again[5].  I don't have time to work through this now, so I'm just recording my meandering thoughts here for future self, and anyone else that would like to take a look.

Hampton writes down the generalization of the law of cosines for polyhedra rather than just the plane, (pun intended), old triangle.  Here it is

$$0 = \sum_j\vec{n}_i\cdot\vec{n}_j\Delta_j = \Delta\left(i\right) - \sum_j c_{ij}\Delta_j$$

Where, $$c_{ij}$$ is the cosine between two faces of the polyhedra i and j, and $$\vec{n}_i$$ is a vector field normal to the i'th face.

Dr. Hampton states that this expression is arrived at through the divergence theorem for polyhedra.

Here's the question, sketchy as it may be.  Since the Legendre polynomials are a solution of Laplace's equation under certain boundary conditions, and since the polynomials can be generated by the law of cosines, in light of the series of cosines above, if we extend the sum out to an infinite number of identical faces for the polyhedra, can we arrive back at the Legendre polynomials?

In addition to his recent article in AMM, Dr. Hampton has produced quite a bit of other material worth checking out [2][3][4].

References:
1.  http://www.jstor.org/stable/10.4169/amer.math.monthly.121.10.937
sadly, behind a pay wall, but see [3]
2.  Marshall Hampton's mathematical coloring book
http://www.d.umn.edu/~mhampton/mathcolor17b.pdf
3.  Marshall Hampton's home page
http://www.d.umn.edu/~mhampton/
4.  Marshall Hampton on Diff Eqs and Sage
https://books.google.com/books?id=78cSvjHo1zUC&lpg=PA233&ots=LZjUB_iA-x&dq=Marshall%20Hampton%20cosines&pg=PP1#v=onepage&q&f=false
5.  Legendre polynomials on Copasetic Flow
http://copaseticflow.blogspot.com/2013/01/law-of-cosines-and-legendre-polynomials.html

### 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…