### 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

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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…