### EM and Complex Analysis

There are an increasing number of apparent correspondences between EM this semester and our section on complex analysis in math methods last semester.  These are just notes on a few of them.

Uniqueness of the Electrostatic Potential Solution and Liouville's theorem
After stating that we would be solving Poisson's equation

to determine electrostatic potentials, our professor then launched into a proof that the solutions, once found, would be unique.  We first defined a potential psi equal to the difference between non-unique solutions, (assuming for the moment in our proof by contradiction that there could be more than one unique solution).  We placed psi back into Poisson's equation and ran through the following steps:

Ultimately we wound up proving that at best psi is a constant, but that it must be zero everywhere on the surface that defines the Dirichlet boundary conditions that the two 'non-unique' solutions both satisfy, so it's constant value must be zero and the two 'non-unique' solutions are in fact the same.

Here's the question.  At the step shown above and highlighted here:

could we have fast tracked the entire proof by invoking a result from complex analysis?  In complex analysis we learned that analytic functions satisfy the condition on psi highlighted above, (Laplace's equation).  We also learned a version of Liousville's theorem that went:

"A function which is analytic for all finite values of z and is bounded everywhere (including infnity) is a constant."

It seems this would have immediately brought us to the conclusion that psi was constant and things could have moved on from there.

Finding Potentials Actually Just Complex Analysis?
There are an accumulation of pointers in my mind that what we're doing when solving for potentials is very closely related to complex analysis and in particular to Cauchy integrals.  When solving for the potential within a volume, we're told that either the value of the potential everywhere on the surface bounding the volume, (Dirichlet boundary conditions), or the value of the normal derivative of the potential everywhere on the boundary, (Neumann boundary conditions), is sufficient information.  This looks a lot like the Cauchy integral idea where if we know the values of a function around a contour, we can calculate the value of the function at any point inside the contour.  Is there anything to this?

Please excuse the obligatory coffee stains.

Picture of the Day:
 From 1/23/13

### 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…