Villanova University Pi Mu Epsilon Induction Talk, May 2, 2014, 9:30am Mendel
Rm 115
by Robert T Jantzen
Inspiration: Stewart 7e-8e Section 10.3 (removed from edition 9e):
Parametric equations and polar
coordinates Figs 16, 17. [How
did this talk come about?].
"Just a figure" but it seemed interesting to push the idea of arclength
integration in section 10.4 while demonstrating the power of Maple (bob is
curious to push math ideas). [This Figure 16 was removed from later versions of the
textbook.]
Scratching on the problem it dawned on bob (eventually, bob was a bit slow here) that there was some interesting stuff to play with here.
Disclaimer: bob has past experience with
pasta geometry. Radial oscillations of the circular cross-section of tube pasta make them "rigati" = ridged, "big" cylinder plus ridges: "rigatoni", ridged corkscrew pasta: "cavatappi rigati"] but that is another story... |
First, why does this seemingly complicated expression for the radial
oscillation look very much like a pure sinusoidal oscillation about the central
circle at the average radius? Answering this question opens a mathematical can
of worms that touches a number of important idea whose natural extension leads
to the recent exciting discovery of the effects of primodial gravitational waves
on the anisotropy of the cosmic microwave background.
[see our own Camille Carlisle's VU '08 astro
major discussion in Sky and Telescope:
http://www.skyandtelescope.com/astronomy-news/direct-evidence-of-big-bang-inflation/
or
Google]
Power reduction formulas: Wiki: trig power reduction formulas [inverse of double, triple, n times angle formulas]
Fourier "harmonic" analysis on the circle generalizes directly to
"spherical harmonic" analysis on the 2-sphere,
for example, used to analyze the celestial sphere
image of the 10^(-5) perturbations of the extremely isotropic 2.725 degree Kelvin
cosmic microwave background (CMB) radiation recently in the news for
apparent confirmation of the inflationary model of the universe:
Maple worksheet: bumpyspheres.mw
improved to bumpyspheres2.mw
[Wiki: Spherical
Harmonics,
CMB images,
CMB power spectrum,
CMB
review]