Simulations & Interactive Learning

Three live widgets, one per arc of the course.

Live

Brachistochrone racer — five paths, five beads, and the cycloid wins every time. The founding problem of the calculus of variations, raced
Conformal map explorer — watch the complex plane bend under \(z^2\), \(1/z\), \(e^z\), the Joukowski map and a Möbius transformation, and find where conformality fails
Contour integrator — residues computed numerically, arcs shown actually vanishing, and real integrals that resist every real method falling in three lines

Widget	Benchmark	Agreement
Brachistochrone	cycloid endpoint solve	\(10^{-16}\)
Brachistochrone	\(T = \sqrt{a/g}\,\theta_1\)	0.02%
Brachistochrone	straight-line ramp \(T = L\sqrt{2/gY}\)	exact
Brachistochrone	cycloid beats all rivals, every endpoint	confirmed
Brachistochrone	tautochrone: \(T = \pi\sqrt{a/g}\), any start height	0.04%
Contour	\(\int dx/(1+x^2) = \pi\), \(/(1+x^4) = \pi/\sqrt2\), \(/(1+x^6) = 2\pi/3\)	\(10^{-13}\)
Contour	\(\int\cos x/(1+x^2)\,dx = \pi/e\)	\(10^{-13}\)
Contour	double-pole residue \(= -i/4a^3\)	\(10^{-11}\)
Contour	arc contribution vanishes as \(R\) grows	monotone

All three arcs of PHY622 are the same move: a strong global constraint forces a rigid local answer.

Variations: requiring stationarity of a functional over all paths forces a differential equation on the extremal.
Complex analysis: requiring differentiability in the complex sense forces Cauchy–Riemann, and thence that boundary values determine the interior.
Groups: requiring closure under composition forces Lagrange's theorem — a subgroup's size must divide the group's, with no computation at all.

Each time, demanding something everywhere pins down what happens at each point.