/sci/ - Science & Math

I'm a dumbass that knows a little bit about real differentiable manifolds and complex analysis, but I came across pic related and I wanna try to decipher what the fuck it means.

Really, the only thing I actually need to understand is how they found the differential of the map they mention (which is the line that's marked near the bottom) so an explanation for that would be good enough. But additionally I'd like to know what I'd need to study to be able to understand the rest of the theory here. Is it just complex differentiable manifolds and some quaternion theory? Also, any book recommendations to get started on this?