Copasetic Flow

Remember when "The New Math" had us all learning set theory in elementary school?  I hated set theory, and had no idea why we were learning it.  This week, years, (ahem, many), year later, I found out what it was all about: Sputnik!  While researching a book in the--as it turns out--aptly named New Mathematical Library, I found the historical link. The NML was a series of books commissioned by the School Math Study Group (SMSG).  A quick dive into Wikipedia turned up the interesting fact that the SMSG was formed after Sputnik flew over.  The US decided we were going to need far more scientists and engineers than we had available.  The solution to the problem back then made logical sense: increase the number of math literate students available.  Hence, the SMSG gave us the NML, a series of excellent books that: "make available to high school students short expository books on various topics not usually covered in the high school curriculum", and the...

Just a few brief notes on circular and hyperbolic trigonometry today.  First, if there's anyone who'd like to offer, any clarifications, expansions, or other cool and interesting facts, please, you're more than welcome!  As usual, this is stuff that I learned in high school that didn't become blindingly clear and meaningful until it cropped up in grad school physics.  First, the equations for the circle and the hyperbola (picture 1) Notice that they differ only by a single negative sign.  Now, for their graphs, (picture 2). Each of these figures can also be expressed in parametric form as follows, (this is where the trigonometric and hyperbolic trig functions come in).  (picture 3) Now for the notes and other thoughts. Angular Arc Length and Angular Area Until just a few weeks ago, I had always wondered why the inverse trig functions were called arcsine, and arccosine.  The answer makes perfect sense, it just hadn't occurred to me.  T...

The linked document below contains an integration 'trick' that's very interesting to me, but it arguably shouldn't be.  First, the interesting bit.  The problem shown is from a derivation of the Fourier transform of the ground state of the hydrogen atom from position space into momentum space  The trick here shows how to easily integrate theta dependency using simple 'u substitution'.  There' a second possibly much more interesting bit that I have to check out later, we wind up with sin(kr)/kr which is a prototype for a delta function, which is related to the whole Fourier transform process, but I digress. While I'm very proud of my u substitution, I arguably shouldn't be.  I'll out myself.  I'm in grad school and I'm not an expert at integrals!  The shame!  After all, I learned the technique of u substitution in freshman calculus. Here's a little bit of history followed by a suggestion I'd appreciate your thoughts on.  When I...