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

4. All about hyperbolic functions
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

More Cowbell! Record Production using Google Forms and Charts

First, the what : This article shows how to embed a new Google Form into any web page. To demonstrate ths, a chart and form that allow blog readers to control the recording levels of each instrument in Blue Oyster Cult's "(Don't Fear) The Reaper" is used. HTML code from the Google version of the form included on this page is shown and the parts that need to be modified are highlighted. Next, the why : Google recently released an e-mail form feature that allows users of Google Documents to create an e-mail a form that automatically places each user's input into an associated spreadsheet. As it turns out, with a little bit of work, the forms that are created by Google Docs can be embedded into any web page. Now, The Goods: Click on the instrument you want turned up, click the submit button and then refresh the page. Through the magic of Google Forms as soon as you click on submit and refresh this web page, the data chart will update immediately. Turn up the:

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics 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

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 sim

Copasetic Flow

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