### Chain Rule Notes

Not too surprisingly, I suppose, a homework problem in EM recently asked us to find the Laplacian in a different coordinate system.  The system of choice this time was parabolic cylindrical coordinates.  At first, I was unconcerned, I had my handy example on how to do this using the line element for the coordinate system[1].  As it turned out, that trick was disallowed by the assignment itself.  We were required to use the chain rule.  The parabolic coordinate system was defined as the set of equations to the left, (picture 1).

The chain rule of calculus for some odd reason has always been my arch-nemesis.  This is particularly problematic for studying physics because we use it to exhaustion.  I'd hoped to provide a more coherent presentation, but more midterms are coming up.  For now I'll just capture the solution for finding the Laplacian with a few notes as to what can be learned, and the motivations for various steps.

Step one
Write down the u and v differentials in terms of x and y abstractly (picture 2):

The above is essentially the multi-variable chain rule.  If you're me, and you can get to this step, the rest is pretty easy.  One thing to notice here.  As physicists, we're often fond of 'cancelling' differentials like fractions, (as a memory aid if for not other reason).  Our mathematician buddies frequently caution us that this is just wrong.  If you 'cancel' the partial x or partial y differentials above and add, you'll see an example of why you have to be careful with this.

Step 2
Calculate the x and y differentials with respect to u and v, (the missing pieces above).  (picture 3)

Step 3
Since we're looking for the Laplacian which involves second derivatives, have faith, press on, and apply 'partial partial u' and 'partial partial v' to the above equations once more.  Next, add the results together (picture 4).

Step 4
I've done a horrible, horrible thing here, one of the things I hate the most.  I've skipped steps somewhat wholesale.  Do be fair, I did them in my head, so I know they can be done, but there you have it.  I did it. (picture 5)

Here's what we're up to.  We want a Laplacian that we can apply to scalar fields in the parabolic cylindrical coordinate system.  If you look at the last set of results above, you'll notice that all the u and v differentials are bound up behind other differential operators.  That won't do.  We need to be able in the end to be able to apply them directly to functions written in the u, v system.  So, we collect our blind faith again and apply 'partial partial u' and 'partial partial v' calculated in step two to the x and y differentials.  After that, all that remains is to collect like terms and make the whole mess look like a Laplacian.  The z term gets tacked on a the end since it was orthogonal to the other two coordinates in each system and had not u or v dependence.

Hopefully, there will be more on this later.

References:

### 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…