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### Rough Cancelling of Zeros and Poles

In electromagnetism, we came across the following formula describing a potential on a plane (picture 1)

In our problem, we were given a plane that looked like the following.  The potential inside the small square is V and the potential everywhere else is on the plane is zero (picture 2)

We were to estimate why the potential on the plane didn't go to zero everywhere when the z in the pre-integral numerator  seemed to indicate that it did. Very roughly speaking, the points within the denominator blow up to infinity and conspire to cancel the numerators 0.  There's a more elegant and rigorous way, (courtesy of my professor), to show that everything is all right though.  Here's a sketch of the technique.

What we want to do is show that when the denominator goes to infinity while the numerator is zero, things cancel nicely and we still get a finite answer.  When the denominator goes to infinity, the following conditions apply (picture 3)

We want to approximate the potential at locations near where these conditions arise.  It's easier to do this using cylindrical coordinates, and we make the following substitutions (picture 4)

For a nice explanation on where the dx-prime times dy-prime element of area substitution came from and for a nice way to think about spherical divergences, see this post[1].  Having made that substitution, the integral above can now be written as (picture 5)

Making the additional substitution w = rho squared we get (picture 6)

This integral can be fairly easily evaluated to get (picture 7)

Where we've taken the integral in the limit where the radius squared goes to zero.  Notice that the first term very nicely cancels the z over 2 pi in the pre-integral factor in the original formula above!

References:
1.  http://copaseticflow.blogspot.com/2012/09/an-intuitive-way-to-spherical-gradient.html

### 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…