Your favorite Eureka moment for proofs

[martinbn]

May be not exactly on topic, but one thing that was an eye opener for me was when I relized that viewing finite sums as integrals with respect to the counting measure can be helpfull. For example in representation theory of finite groups. Even the definition of the group algebra as the space of all complex valued function on the group with product convolution is for me much clearer conceptually.

 

[king vitamin]

It was a very esoteric and specific result, but my biggest Eureka moment as a physicist so far was my discovering the proof I published in https://arxiv.org/abs/1610.06568, the short proof given in Section II.B. I didn't believe it when I first wrote it down because it was such a simple and seemingly straight-forward result, whereas all my other results I've discovered in my research were slow and difficult slogs which I understood over an extended amount of time. But this was literally a case of looking at some equations on a chalkboard and realizing how they all worked out to give a simple and powerful result - and we immediately checked that my general result agreed with several previous special cases. It was a very satisfying moment compared to most of my research which is often long and difficult calculations!

 

[Charles Link]

My biggest Eureka moment was in June 2010, when I was doing a test case to try to show consistency/inconsistency between the magnetic pole model and the surface current model of magnetostatics. I was using a uniformly magnetized cylinder of arbitrary radius $a$ of semi-infinite length. The pole model gives a very simple result of $B_z=0$ for the z component of the magnetic field in the plane of the endface, for $r>a$, everywhere in the plane, (because the only pole is a uniform magnetic surface charge $\sigma_m=M$ that sits on the endface). I thought it very unlikely that the Biot-Savart integral of the magnetic surface currents on the outer surface of the cylinder (of semi-infinite length) could possibly give this $B_z=0$ result for $r>a$, but they did, and they also gave the correct $B_z=2 \pi M$ (cgs units) for $r<a$ in the plane of the endface. I was very pleasantly surprised. The two models were completely consistent. With a little extra logic, I was able to prove the pole model formula $B=H+4 \pi M$ follows from this Biot-Savart/surface current result.

 

[jedishrfu]

Sometimes a Eureka moment comes following someone’s comment. We were doing a tough classical mechanics problem from Marions book. We were to prove the a particle falling from outer space would take 9/11 the total time to fall half the distance. (Marion/Thornton Chapter 5 problem 5.5 pg 205

Newton’s equation had an r and would give us the acceleration but we couldn't relate the r to the time. We mulled over it for quite awhile until another prof came in and suggested Kepler‘s law of equal areas in equal times.

It was then that we realized that we could make an orbit and collapse one axis and then we had the missing connection of r to t.

 

[DaveE]

For me, I think it was when I understood the δ-ε proofs in calculus. Somehow, I wasn't impressed with the Limits I'd learned shortly before. This was the first time I understood that this could be a really useful and different approach. "Arbitrarily close" wasn't a concept I'd known before.

 

[ChinleShale]

For me, I think it was when I understood the δ-ε proofs in calculus. Somehow, I wasn't impressed with the Limits I'd learned shortly before. This was the first time I understood that this could be a really useful and different approach. "Arbitrarily close" wasn't a concept I'd known before.

These proofs are fundamental for understanding a lot of mathematics.

 

[ChinleShale]

To me there are at least three types of Eureka moments. One is when one understands a proof of a known theorem. A second is when one tries to prove a known theorem on one's own. The third is when one tries to discover a new theorem. This third one can take the form of seeing a relationship that unifies different examples. In math this Eureka moment often does not take the form of a rigorous proof but more of an insight. The rigorous proof comes later.

This last type is not unlike the process of discovering a new theory in any scientific field. @king vitamin 's post #32 describes this. In my own experience I was once studying a conjecture about the groups of isometries of a class of manifolds. I spent a year trying to construct a counter example. Reams of newsprint later, I saw a relationship that explained why no counter example exists. It took a lot of effort afterwards to find a rigorous proof.

 

[songoku]

My eureka moment happens every week here in PF homework section when I finally understand the hint given in post #2 of my thread after the helpers give another 50 hints