### 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
http://www.d.umn.edu/~mhampton/
4.  Marshall Hampton on Diff Eqs and Sage
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

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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 differe…

### Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…