Favorite equation

18 May 2021

The divergence theorem, also referred to with any combination of names "Gauss", "Green", and "Ostrogradski".

$$ \int\limits_V \nabla \cdot \vec{F} \hspace{5pt} dV = \int\limits_{\partial V} \vec{F} \cdot d\vec{s} $$

It's one of those equations where it looks like such a cryptic, arbitrary relation, but once you get it you know that it must obviously be true. Like $1+1=2$ level of true. Arguably the generalized Stokes theorem is prettier, but not as satisfying.

$$ \int\limits_\Omega d\omega = \int\limits_{\partial\Omega} \omega $$

Show this to anyone and they will scratch their head wondering what does the $\int$ sign even mean without some sort of $dx$ to go with it (hint: it's hidden in the $\omega$). Not nearly as intuitive.

Close second would be the Born rule (Born postulate if you are a communist), which is the very reason everyone saying: "Physicists don't use Akham's Razor, just look at quantum mechanics!" is wrong and cannot be trusted.

$$ \mathbb{P}(\lambda_n) = \vert \langle n \vert \psi \rangle \vert^2 $$

Just look at it. So pretty. And so intuitive.