### Springtime in Colorado and Ice Formation

We're on the road today, so I may not get the chance to do a full post.  Elaine mentioned the other day that it had been awhile since I'd put any pictures up here, so for today's short, but hopefully pretty and interesting post, here are the lake ice formations from Lefthand Reservoir near Ward, CO.  These are springtime pictures taken in March of 2010 and the lake ice was at least a foot thick.  Towards  the end, you'll find a video of Maya the swimming super dog as a sweetener!  Don't worry, she's not swimming in the ice water, she's in Long Island Sound near Sound Beach, NY, a mere four miles from Wardenclyffe and ten or so miles form Brookhaven National Laboratory.

First, the setting.  Here's a topo map of the lake.  It sits at an altitude of about 10,600 feet roughly, 40 miles outside of Boulder, CO and immediately outside of Ward, CO.

View Larger Map

The lake itself and it's surroundings are gorgeous.  Here's a little sample.

Now for the somewhat sciencey stuff.  The last time we were up there, we found two points in the ice where holes had been made and then refrozen.  In the first hole, there were what I believe were ice crystals that had formed.  You can actually see several inches down the crystal into the lake!  In the second hole there were a series of small bubbles frozen inside.  Here's are a few pictures.  A few videos that let you get a little side to side perspective on the geometry follow.  Does anyone have any insight into how these kinds of features are formed?

 Ice Crystals
 Ice Bubbles
 Ice Bubbles with Knife for Scale
Ice Crystal Movie

Ice Bubble Movie

Maya the Super Duck Dog!

### 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…