They are supposed to hit the curvy bits at the midpoints of the intervals.
the curve is the absolute value of the Dirichlet kernel $$|D_N(t)| = \left| \frac{\sin\left(\left(N+\frac{1}{2}\right)t\right)}{\sin\left(\frac{t}{2}\right)} \right|$$
@JakeRose What do you mean by stationary points? As far as I can tell, $f$ maps $\mathbb{R}^2$ into $\mathbb{R}$... those aren't even the same spaces...
There's a numberphile video on it, but it's because someone somehow collaborated on a paper but was added into the authors after the paper was published or something, I can't quite remember
anyway that persons Erdos number was given as something like $1 + i$
Okay so I only really know how to execute things in forms, so we want $\psi \nabla \phi \cdot dS$ is given by a form whose exterior derivative is $(\psi \Delta \phi + \nabla \psi \cdot \nabla \psi) dV$
You should figure out how to translate to forms. If $\vec F$ corresponds to the $1$-form $\omega$, then the flux of $\vec F$ across the boundary is given by integrating $\star\omega$ on the boundary.
There have been books I've read where, after proving several lemmas, just write: "Theorem: Blah blah. Proof: By applying Lemma 1 to the result of Lemma 2, the result follows." @LeakyNun
Okay @Ted I'm back and with TeX reading abilities. So, I'll just expand things out I guess. Let's say we have $U\subset \mathbb{R}^n$, and the functions are $\psi,\phi:U\to\mathbb{R}$. Then $\nabla \psi \cdot \nabla \phi = \sum_{i=1}^n \psi_i \phi_i$
@HsMjstyMstdn It's a strange fact that if you add together a sine wave (of some amplitude and phase) plus another sine wave (of some other amplitude and phase) you get another sine wave, rather than anything weird
If you add together two sine waves of different periods (aka different frequencies aka different wavelengths), though, the result is no longer a sine wave
$\sin(x)+\sin(3x)$ does not look like a sine wave
but as long as you only change the phase and amplitude it works
So the way I know it is this, I'm taking $dV$ as the volume form and if you have some vector field $F = (F_1,\ldots,F_n)$, you take the form $\omega = \sum_{i=1}^n F_idx_i$, so that $\int_{\partial U} F\cdot n dS$ (whatever that means) is $\int_{\partial U} \star \omega$
