Determining Whether Voltage Lags or Leads Current in an RLC Circuit Extra Class Exam Style

If you're coming in from the physics side
In my spare time I write web apps. I also play with ham radios as KD0FNR.  One of the web apps I've published provides practice exams for ham radio license tests.  What follows is a study video for the extra class exam.  The extra class license is the highest class of license an amateur operator can earn in the United States and the test requires quite a bit of electronics theory.

If you're coming in from the ham radio side
For my full time gig, I'm a graduate student in the physics department at Texas A&M University.  In my spare time, I write about physics topics that catch my eye on any given day.

Now for the study video
This study topic details how to work through the questions on the exam that ask about the phase relation between voltage and current in a series RLC circuit.  It shows a few tricks for doing the problems without a calculator.  All feedback is welcome.. please!  As for me, I feel this video is a bit flat, (there aren't any pirates or a moose), but I believe it gets the basic points across.  #E_5_B_07

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…