tan(x)=x The Overlooked Approximation and Other Notes

My quantum professor always knows exactly the right approximations to make, (I suppose it helps to be the person who wrote the homework). Regarding the recent homework questions involving boundary conditions in shallow square potential wells which are just oozing with tangents of ratios of wave numbers, he pointed out a trig approximation that's often overlooked, (at least by students like myself). At small angles (picture 1)

the tangent of x is just x.

Yup, small angles aren't just for sine anymore! The tangent of a small angle is also equal to it's argument. I had to go through the additional little thought process of (picture 2)

Here's a graph of tan and sin overlayed with x so you can see what's going on. (picture 3)

The Levi-Civita Symbol
Another mathematical tool that has come up repeatedly in the last few weeks is the Levi-Civita symbol. If you've studied the cross product or the vector curl you've seen it. Wikipedia defines the symbol like this (picture 4)

They go on to say that it's +1 if the indices are an even permutation of (1,2,3), and -1 if the indices are an odd permutation of (1,2,3). My rule for remembering this is that if you can read the indices in order from left to right, (wrapping when you hit a close paren), then the results is +1. If you can read the indices in order from right to left, (wrapping when you hit an open paren), then the result is -1. Positive numbers on the number line go from left to right and negative numbers go from right to left... Voila.

This symbol turns up over and over again and is very handy to memorize. When you want to put things succinctly, and impress your friends, you can write down the components of a cross product, or a curl, or any number of electromagnetics equations using the Levi-Civita symbol. For example, Wikipedia defines the vector curl as (picture 5)

References:
1. http://en.wikipedia.org/wiki/Levi_civita_symbol

2. xkcd style graphs:
http://mathematica.stackexchange.com/questions/11350/xkcd-style-graphs

Picture of the Day:
Today's pictue of the day is taken from Wikipedia and is of Tullio Levi-Civita. He's one of the few 19th and 20th century scientists I've researched lately who was actually smiling! He's cool! (picture 6)

Comments

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

Search This Blog