the hidden math of a polar coordinate integration problem

Villanova University Pi Mu Epsilon Induction Talk, May 2, 2014, 9:30am Mendel
Rm 115

by Robert T Jantzen

*r *= sin^{2}(2.4 θ) + cos^{4}(2.4 θ)

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]

- via Fourier series analysis (Fourier polynomials): Maple worksheet fourierpowers.mw
- via complex analysis and the binomial theorem: Maple worksheet: spaghetticircle.mw (see section: complex analysis)
- But these two join together in the complex exponential Fourier series
analysis!

- harmonic analysis on the circle (or a fixed closed interval = guitar
string, wind instrument, etc: tone, overtones)

[see Wiki "vibrating strings"]

(another intriguing figure from his calc book)

**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]