/sci/ - Science & Math

>>11778249
lel

>>11778247
>The equation states that the gradient of the numerator is collinear with the gradient of the denominator.
Which sounds like it ought to be a constant while it isn't.

This reminds me of counterexamples to holomorphic but naively nice looking functions (althought this isn't in the domain of several complex variables). One would hope
[math] f(z):=\overline{z} [/math]
to have a somewhat constant derivative, but in fact
[math] \dfrac{f(z)-f(0)}{z-0}=\dfrac{\overline{z}}{z}={\mathrm e}^{-2i\,\mathrm{arg}(z)} [/math]
(and the same is thus true for Re and Im as functions)
The thing is constant along rays, but even when f goes to 0 when z becomes smaller, the derivative fluctuates arbitrarily along the angle direction.

I can't help more with just that one page, but maybe that helps somehow.