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Aug
27
awarded  Notable Question
Aug
26
comment Prove the differential of $f$ at $p$ is a well defined linear map
Yes, I think it's correct and it is (essentially) a consequence of the chain rule.
Aug
17
comment Closed orientable three manifold with finite cover by $S^1 \times S^2$ or $T^3$
The only orientable $3$-manifolds finitely covered by $\mathbb{S}^1\times \mathbb{S}^2$ are $\mathbb{S}^1\times\mathbb{S}^2$ itself and $\mathbb{RP}^3\# \mathbb{RP}^3$.
Aug
11
comment Proving that the limit of $\frac{x^{2}-1}{x-1}$ as $x \rightarrow 1$ is $2$
If $x\neq 1$, then $\frac{x^2-1}{x-1} = x+1$. All that matters when evaluating the limit $\lim_{x\to 1} \frac{x^2-1}{x-1}$ (and writing down an $\epsilon-\delta$ proof) are the values of the function $\frac{x^2-1}{x-1}$ when $x\neq 1$.
Jul
9
comment Several statements about $\mathbb{R}$ with chart defined by $f(x)=x^3$
... If you think of a chart as being a map in the other direction (i.e., $f:X\to\mathbb{R}$ is defined by the rule $f(x)=x^3$), then $\phi(x)= x^3$ would be correct (because now verifying that $\phi:X\to\mathbb{R}$ is a diffeomorphism is tantamount to verifying that $\phi\circ f^{-1}:\mathbb{R}\to\mathbb{R}$ is a diffeomorphism, which is now so since this is the identity (but would be $x^{1/9}$ with $\phi(x) = x^{1/3}$). So, I don't think this is a mistake (for either of us).
Jul
9
comment Several statements about $\mathbb{R}$ with chart defined by $f(x)=x^3$
Hi @Vadim, thanks for your comment. I guess this is simply a notational point depending on how you define a "chart". In my answer, I think of the chart $f(x) = x^3$ for $X$ as being a map $\mathbb{R}\to X$; with this definition, verifying that $\phi:X\to \mathbb{R}$ is a diffeomorphism is tantamount to verifying that $\phi\circ f: \mathbb{R}\to \mathbb{R}$ is a diffeomorphism; of course, this is so because the composition is the identity map of $\mathbb{R}$ ...
Jul
8
comment How to study math to really understand it and have a healthy lifestyle with free time?
Hi @JPG, please see my above four comments where I provide a more elaborate response to your question.
Jul
8
comment How to study math to really understand it and have a healthy lifestyle with free time?
In the long run, I managed to cover the basic topics, and this suited me better than "systematically" working my way through mathematics. In principle, I wouldn't recommend this to a beginning mathematics student (in fact, I may not even recommend it to a younger version of myself) but that's the way I learnt, at least earlier on. The main suggestion that I can honestly give is to be as creative as you can and find your own way through mathematics - and then give advice to other people based on what you've learnt! Sometimes the only way to learn from some mistakes is to make them yourself.
Jul
8
comment How to study math to really understand it and have a healthy lifestyle with free time?
Ultimately, how one learns math really depends on many factors such as how experienced one is, how mathematically mature one is, how much time one has, how one is able to learn new things etc. so it is hard to give a really specific answer. I mean one thing that worked well for me early one was not to necessarily learn things in order but rather as I saw interesting. For example, I studied topology (point-set and basic algebraic topology) without having really learnt linear algebra, real analysis and complex analysis; I sort of just looked up some things in real analysis as I needed them ...
Jul
8
comment How to study math to really understand it and have a healthy lifestyle with free time?
Also, proving results on your own definitely helps to be actively involved with the subject, and is a great way (I think) to test one's understanding early on (it was a slight jump for me when I first started reading serious math textbooks after calculus, because solutions to exercises weren't present, so that's why I initially started taking the theorems as exercises with solutions). However, in the long run, it can be time consuming to prove everything (or even many non-trivial things) on one's own; it's a tricky balance. (Also, some things appear trivial if you have the right perspective.)
Jul
8
comment How to study math to really understand it and have a healthy lifestyle with free time?
Hi @Pacerier, thanks for your comment; that's a good point. I think it is fair to say that it wasn't the best approach for me to take so many notes, in hindsight. It didn't really help me to remember the mathematics any better; I think it is better not to worry too much about forgetting things, and just to be at peace with it. As one gets more mathematically experienced, the really important ideas will be clear as well as a broad(er) vision of what certain aspects of the subject are about (details of proofs etc. are sometimes best forgotten) ...
Jul
8
comment Several statements about $\mathbb{R}$ with chart defined by $f(x)=x^3$
Hi @Vadim, I used the "atlas definition" for smooth manifolds and smooth maps, which is equivalent to the "functional structure definition". I like the functional structure definition a lot, perhaps more than the atlas definition, but I think the atlas definition is the most commonly used definition (e.g., in books on differential geometry). For example, it is probably discussed in Wikipedia.
Jul
8
answered Several statements about $\mathbb{R}$ with chart defined by $f(x)=x^3$
Jun
28
comment How to study math to really understand it and have a healthy lifestyle with free time?
Hi @JPG, thanks for your comment! It's hard for me to recommend anything without knowing your specific situation but I think the main thing is to think about mathematics in your own way, in addition to what you are learning. Of course, there are many ways of doing this and what I suggested is just one way. It's worth trying various approaches and then deciding which ones work for you.
Jun
27
comment How to study math to really understand it and have a healthy lifestyle with free time?
Hi @Pacerier, I wrote them in several notebooks by hand, and they are probably somewhere in my house. Perhaps I will try to find them now that you mention it!
Jun
22
comment How to study math to really understand it and have a healthy lifestyle with free time?
Hi @Pacerier, thanks for your comment. I've never actually gone back and re-read my notes (though I would be curious to do so at some point, even if only to see what I was thinking the first time I was learning the subject). If I do review a subject, then for the reason you mention, I would go back and re-read the relevant parts of the book (which would probably be better exposition then my rough notes, anyway). Nowadays, I don't adopt this learning style; it takes too much time and doesn't necessarily help me to remember/understand the subject better!
Jun
18
comment Show that there is no analytic bijection from the unit disc to $\mathbb{C}$
Hi @Aloysius, if I understand your question correctly: "set-theoretically", the map admits an inverse because it is bijective. Now, we know that locally this inverse map is holomorphic, so it must therefore be (globally) holomorphic.
Jun
18
comment Show that there is no analytic bijection from the unit disc to $\mathbb{C}$
I guess most complex analysis textbooks should contain this fact. However, here is a proof sketch ($\phi$ is holomorphic near $0$ and WLOG $\phi(0)=0$): if $\phi'(0)\neq 0$, then $\phi$ admits an analytic inverse near $0$; otherwise, $\phi$ has a zero of order $n>1$ at $0$ and $\phi(z)=z^n\psi(z)$ for some $\psi$ which does not vanish near $0$. Locally, $\psi$ thus has an $n$th root (say, $\eta$) so we can write $\phi(z)=(z\eta(z))^n$ which is clearly not injective (it is $n$-to-$1$). I've possibly used something non-trivial in this proof, so let me know if I should elaborate.
Jun
18
comment Show that there is no analytic bijection from the unit disc to $\mathbb{C}$
Hi @Aloysius, it's a special property of holomorphic functions that a bijective holomorphic function admits a holomorphic inverse (an analogous statement is not true for real analytic functions as $f(x)=x^3$ shows). The key fact needed is that an injective holomorphic function must have non-vanishing derivative. Are you aware of this?
Jun
13
comment About the topology textbook.
I second the suggestion of @alkabary of using Munkres (this was the book that furnished my first introduction to topology).