/sci/ - Science & Math

>>11077278
>that rely only on set theory.
If you use synthetic axiom, why the fuck do you need sets?

>>11077171
How do you spot me, scary

>>11077162
Pretty sure this is about representations of matrices over Z

>>11071784
>>11071784
Let (x,y)=z denote the real and imaginary coordinate and z the 2d vector.
Allow me to use the same name f:C->C for the complex function and the vector field, i.e. I'll write f(z) and also f(x,y).

If a function f is analytic on an open complex domain, then it's complex derivative at a point [math]z_0[math] is the limit [math] \lim_{z\to 0} \dfrac{f(z_0+z)-f(z_0)}{z-z_0} [/math]. This is a more restrictive notion than just the Jacobian of f(x,y), which is component wise.
In terms of the latter, the former requires the Jacobian matrices in the domain all to be a representation of R. Those are the Cauchy Riemann differential equations (the imaginary derivative is already determined by the real one.)

So instead of just requiring something about the real component in the direction of the really coordinate, the analyticalally of f actually determines how the function can vary in the imaginary direction. That is to say, you can cross the boundary of the domain in only one way.

>>10922434
You need linear transformations and it's what you learn in the book. But the book of Artin is the the theory of linear transformations what a youtube video history lesson is to studying history. (Best example I could come up with on the fly. Point being it's a vast topic.)

I try to push the "reading club" (unless in the next 3 days nobody else is posting) and I happen to work in computer vision. Pic related is a standard text that covers the various relevant transformations in the field early on and readings such as Artin will indeed by very helpful to understand that book