### Notes on Emmy Noether and Group Theory

Just a few quick notes on historical trails I'm finding as I study group theory and Emmy Noether's theorem. The people involved are a bit of a group themselves. The first person I found is Lagrange. He introduced the Langrangian, one of the key concepts of analytical mechanics. It's used today, well, everywhere, from plain old mechanics to quantum mechanics, to quantum field theory. He also laid some of the foundations of group theory working on permutation groups and their use in solving polynomials.

That brings us to Évariste Galois. Galois read Lagrange's papers at the age of 15. He later went on to develop Galois theory. Galois theory relates permutation groups of the roots of polynomials to their solvability.

Sophus Lie developed Lie groups, provide a framework that is similar to Galois theory for studying the symmetries of differential equations.

And that brings us to Emmy Noether. Her paper on "Invariant Variation Problems" applied Lie's work to variational calculus revealing conservation laws applied to Lagrangians.

### 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…