### The Care and Feeding of anti-Gudermannians: Spotting Them in the Wild

I mentioned yesterday that the anti-Gudermannian had come up in several articles I’d read, but that the authors hadn’t recognized, or pointed out the anti-Gudermannians lurking in their formulas.  This is a brief set of instructions on how to recognize anti-Gudermannians in their natural state which can be pretty  messy looking.  The first thing you’ll need to know how to spot the multiplicative inverse of something known as the quotient function[5].  It looks like this,

but may be frequently disguised as

The quotient function is very interesting in its own right and turns up all over in things like electrical transmission line formulas, quantum mechanical transmission and reflection coefficients, and in optically active material formulas.  For complete coverage of the quotient function, see Lindell in AJP[5], (sorry I couldn’t find an open access version, so you’ll have to head to your closest university library).  As cool as it is though, it’s only the start of the Gudermannian.

Next, you’ll want to look for a hyperbolic arctangent.  They often look more like quotient functions than hyperbolic arctangents:

Finally, if x in the above formulas represents the sin of an angle, then you’ve found yourself an anti-Gudermannian[3, 4 p. 14]:

The form shown above is how the anti-Gudermannian appeared in the research I’ve been studying.  In MacColl’s article on relativistic projectiles in gravitational fields, we have:

which shows how to find the angle for the maximum range using an anti-Gudermanian.  In Sarafian[7], it turns up looking very disguised as:

The expression inside the natural log can be massaged into a the proper quotient function.  Rather than do the math here, I’ll just reference another article by Dou and Staples[8] which has an expression for the arc length that is more immediately suggestive:

One last little note.  The angle found by both of these formulas, relativistic range, and classical arc length, (56.46 degrees), is also the maximum angle that a catenoid soap film can make with the wall of its containing cylinder, (imagine a horizontal cylinder between the rings in the picture shown below), before it will become unstable[9].  Catenoid’s are surfaces of rotation constructed with catenaries.

References:
1. Mercator Projection
https://en.wikipedia.org/wiki/Mercator_projection

2.  Hyperbolic functions
https://en.wikipedia.org/wiki/Hyperbolic_function#Inverse_functions_as_logarithms

3.  Gudermannian
https://en.wikipedia.org/wiki/Gudermannian

https://archive.org/details/hyperbolicfuncti020206mbp

5.  Quotient functions
http://dx.doi.org/10.1119/1.18845

6.  MacColl on Relativistic Projectiles in +Mathematical Association of America's AMM
http://www.jstor.org/stable/2302436

6.a.  More of interest on MacColl
http://publikationen.ub.uni-frankfurt.de/frontdoor/deliver/index/docId/2678/file/flow.pdf

7.  Sarafian on projectile motion
http://dx.doi.org/10.1119/1.880184

7.a.  More Sarafian on the Angles of Parabolic Trajectories
http://www.mathematica-journal.com/issue/v9i2/contents/MagicAngles/MagicAngles.pdf

8.  Dou and Staples
http://www.jstor.org/stable/2687203

9.  Ito and Sato on Catenoids
http://arxiv.org/pdf/0711.3256v5.pdf

### 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…

### Lost Phone

We were incredibly lucky to have both been in university settings when our kids were born.  When No. 1 arrived, we were both still grad students.  Not long after No. 2 arrived, (about 10 days to be exact), mom-person defended her dissertation and gained the appellation prependage Dr.

While there are lots of perks attendant to grad school, not the least of them phenomenal health insurance, that’s not the one that’s come to mind for me just now.  The one I’m most grateful for at the moment with respect to our kids was the opportunities for sheer independence.  Most days, we’d meet for lunch on the quad of whatever university we were hanging out at at the time, (physics research requires a bit of travel), to eat lunch.  During those lunches, the kids could crawl, toddle, or jog off into the distance.  There were no roads, and therefore no cars.  And, I realize now with a certain wistful bliss I had no knowledge of at the time, there were also very few people at hand that new what a baby…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…