If I understand correctly, Gauss proved that given any oriented Riemannian surface, one can find a complex structure on the surface so that the metric on the charts is just $f|dz|$, where $f>0$. I’ve heard these coordinates referred as “conformal coordinates,” which makes sense, but I’ve also heard of them referred to as “isothermal coordinates.” […]

Taylor series for $\sin(x)$ and $\cos(x)$ around $0$ are well known and usually exercised in beginner calculus courses. However as we know Taylor series are very localized, ultimately fitting only one point and a series of limits of that one point (differentials of integer orders): $$\begin{align}\sin(x) &= \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{(2k+1)!} \\\cos(x) &= \sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(2k)!}\end{align}$$ We also […]

Background. Let $x$ denote an arbitrary real number. Then $x^n$ can be defined for each $n \in \mathbb{N}$ as follows: $$x^n = \underbrace{x \times \cdots \times x}_n$$ If $x$ is furthermore non-zero, then we can extend the above definition by declaring that $x^k$ makes sense for each $k \in \mathbb{Z},$ by defining: $$x^{-n} = \underbrace{(1/x) […]

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been trying to build a quaternion extension of $\mathbb{Q}$. I have been pursuing an overly optimistic method, which has not quite been […]

I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don’t know what does it really mean and what differences makes it to be more ‘rigorous’. My understanding of a proof is that a proof is some explanation to convincing others that a statement is […]

If mathematics is the science of patterns, what pattern does set theory study? My thoughts so far: set theory studies the pattern of relationships between members of a collection and between collections. Background: This isn’t a homework question, I’m starting on Introduction to Mathematical Thinking and ensuring I have a good grounding in the fundamentals […]

If $a \in S$ is some invertible element in a ring $S$, then a computation shows $$\pmatrix{a & 0 \\ 0 & a^{-1}} = \pmatrix{1 & a \\ 0 & 1} \pmatrix{1 & 0 \\ -a^{-1} & 1} \pmatrix{1 & a \\ 0 & 1} \pmatrix{0 & -1 \\ 1 & 0}.$$ If $R \to […]

I haven’t taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat’s little theorem, the law of quadratic reciprocity, etc. rely upon techniques found in abstract algebra? Thanks.

I want to ask if there is some book that treats Differential Geometry without assuming that the reader knows General Topology. Well, many would say: “oh, but what’s the problem ? First learn General Topology, and you’ll understand Differential Geometry even better!” and I agree with that, but my point is: I’m a student of […]

I notice some problems has many different proofs, do all theorems have multiple proofs, is there some theorems which has only 1 way to prove it? $n$ ways? infinite?

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