Clouds Eats Mountain! Film at 7

I came across this picture of Sierra Blanca near Ruidoso, NM last night.  The peak of Sierra Blanca sits at 11,981 feet, above sea level more than 4,000 feet above Alto, NM where the picture was taken.  As a kid, growing up in Ruidoso, we'd say things like, "Wow, that cloud just hunkered down on top of the mountain." just assuming the cloud had come from somewhere else.  Looking at how the cloud above seemed to be literally streaming into the peak, I wondered if there was some kind of sciencey correlation between clouds and mountains.  I found the answer in an article from the February, 1901 issue of "The School World" by George Chisholm[1] that explained mountains actually help to form clouds, not attract them from afar.

Here's how it works.  Wind carries air saturated with water vapor into the mountain where it is forced up the slope.  As the air rises it expands due the lower air pressure, and as it expands, it becomes colder.  As it cools, the temperature of the air can drop below the saturation point for water vapor causing the vapor to condense into water droplets and it is these water droplets that actually form the cloud.  Mr. Chisholm points out
"It is very important however, that boys and girls should at least get a firm hold of the fact that air (like other gases) cools in expanding, even though it is not necessary in teaching geography to explain how this effect is brought about.  It is not very easy to illustrate this effect in a class, but it may serve to impress the fact on the minds of boys and girls to let them know that some freezing machines are based on this principle, and that all those who cycle may observe for themselves the opposite effect, that of heating by compression, by feeling the bottom their inflator when pumping up their tyres."
Here's a closer look at the streams coming off the peak.

References:
1.  On cloud formation and other geography test errors

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…