Anonymous at Tue, 4 Jun 2024 12:26:15 UTC No. 16209661

>>16209629
Riemann is looking swag in that picture.

Anonymous at Tue, 4 Jun 2024 14:04:01 UTC No. 16209784

this is my favourite, Russians in general have the best books on these subjects in my opinion
also if you can read french this title is also great: Analyse et Algèbre - Cours de mathématiques de deuxième année avec exercices corrigés et illustrations avec Mapple by Stéphane Balac and Laurent Chupin
there is also a series of 2 books from oxford that seem to be extensive on the topic :
1. Multidimensional Real Analysis I: Differentiation (Cambridge Studies in Advanced Mathematics, Series Number 86)
2. Multidimensional Real Analysis II: Integration (Cambridge Studies in Advanced Mathematics, Series Number 87)

Anonymous at Tue, 4 Jun 2024 14:44:10 UTC No. 16209831

I forgot to add that analysis is actually intended to be a dense subject, t doesn't omit any part of the building theoretical part

Anonymous at Tue, 4 Jun 2024 14:47:42 UTC No. 16209837

>>16209784
Thank you, I've heard about Zorich. Is it just calculus with proofs, or does it make connections with more advanced topics, like topology?

Anonymous at Tue, 4 Jun 2024 15:09:24 UTC No. 16209871

>>16209837
yeah I remember that it foreshadows things like the Lebesgue integral being an exercise on the first volume. multiple definitions of the same thing are offered, giving multiple points of view to the same concept

Anonymous at Tue, 4 Jun 2024 15:12:23 UTC No. 16209875

>>16209837
and since you mentioned topology, yes most of the time the book is about sets, their properties and what happens to them under mappings. instead of the calculus approach of mappings being the center of attention

Anonymous at Tue, 4 Jun 2024 15:20:44 UTC No. 16209886

>>16209837
also yeah, the definitions are using the topological way. for example a compact isn't defined to be a closed and bounded subset of R^n, that's a theorem. instead the general definitions are proposed, which give where the idea of the notion was taken. like here compacts are sets that any sequence mapped to them has a subsequence that converges (which makes it easy to understand that Q the rational number and its' subsets aren't compacts)
generally if you can comfortably download books from libgen I suggest having many books, the math sorcerer gave this advice too

Anonymous at Tue, 4 Jun 2024 22:56:50 UTC No. 16210649

I asked this question in another thread but I think it’s pertinent here too. For someone who hasn’t done rigorous single variable calculus, can he just do rigorous multivariable calculus/analysis and just pretend n = 1 when proving theorems in R^n? I mean, if something holds in any dimension, it definitely should hold for R.

Anonymous at Wed, 5 Jun 2024 01:10:30 UTC No. 16210843

>>16210649
if you download both volumes of Zorich, you can consult both version of every definition/theorem and compare and fill the gaps imo

Anonymous at Wed, 5 Jun 2024 07:53:21 UTC No. 16211189

>>16210649
Likey not; many theorems in R^n use theorems in R in their proofs, and follow similar ideas.

Anonymous at Wed, 5 Jun 2024 08:21:16 UTC No. 16211205

>>16210649
No. The difference between 1 dimension and more than one dimension is very significant. Not everything "carries over", the intuitions are much different, etc.