So chapter 10 of Pirani and Crampin is where it all comes together. I haven't studied it properly yet but did read through it quickly - it basically goes over most of the work done in the first 9 chapters of the book and puts it into the context of an abstract differentiable manifold, rather than either an affine space or an embedded surface in 3D space. And the results are all the same! There's a bit of abstraction and technical detail using bump functions to justify this, but in reality and from a calculation perspective I don't think there's anything new. It's a good revision chapter.
Before taking up the baton of differential geometry again this week I all but finished the analytic number theory course, completing the proof of the prime number theorem. Feels like an achievement. I've understood evey bit and followed all the proofs, but if I was asked to write it out from memory - I'd have no chance!
Manifolds, prime number theorem
So chapter 10 of Pirani and Crampin is where it all comes together. I haven't studied it properly yet but did read through it quickly - it basically goes over most of the work done in the first 9 chapters of the book and puts it into the context of an abstract differentiable manifold, rather than either an affine space or an embedded surface in 3D space. And the results are all the same! There's a bit of abstraction and technical detail using bump functions to justify this, but in reality and from a calculation perspective I don't think there's anything new. It's a good revision chapter.
Before taking up the baton of differential geometry again this week I all but finished the analytic number theory course, completing the proof of the prime number theorem. Feels like an achievement. I've understood evey bit and followed all the proofs, but if I was asked to write it out from memory - I'd have no chance!