### Use the Calculus Luke!

Yeah for Google+!!!  After posting yesterday's missive about figuring out the origin of the secant squared term, mathematician extraordinare and Google+er +John Baez pointed out to me that the derivative of tangent is secant squared and that I need not have fussed so much with the geometry.  John said:
I wouldn't  call the introduction of sec^2(θ) a "substitution".  Introducing θ in the first place counts as a substitution, since you're trying to simplify an integral by replacing some other variable with this one.   But it's just a mathematical fact that the derivative of tanθ is sec^2(θ), so
d tan(θ) = sec^2(θ) dθ
I would use this equation automatically and unthinkingly, but it looks like you're trying to find a geometrical explanation of this equation.  If you're figuring it out for yourself because nobody told you the derivative of tan(θ) is sec^2(θ), that's very laudable!  The standard approach is to express tan in terms of sin and cos, and use the quotient rule for derivatives.﻿

I guess the lesson is when you see differential on both sides of an expression, think calculus, not geometry!

### 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…