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1500 questions
113
votes
3 answers
Is the derivative the natural logarithm of the left-shift?
(Disclaimer: I'm a high school student, and my knowledge of mathematics extends only to some elementary high school calculus. I don't know if what I'm about to do is valid mathematics.) I noticed something really neat the other day. Suppose we…
David Zhang
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113
votes
15 answers
Intuitive explanation of Cauchy's Integral Formula in Complex Analysis
There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive direction, then for every $z$ in the disc we can…
Pandora
- 6,934
113
votes
0 answers
Classification of local Artin (commutative) rings which are finite over an algebraically closed field
A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(\mathcal{A})\rightarrow X$ where $\mathcal A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ is an infinitesimal thickening…
user16544
113
votes
3 answers
Direct proof that the wedge product preserves integral cohomology classes?
Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$. There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients in $\mathbb R$, which is isomorphic to the De Rham…
Greg Graviton
- 5,454
113
votes
41 answers
Theorems with an extraordinary exception or a small number of sporadic exceptions
The Whitney graph isomorphism theorem gives an example of an extraordinary exception: a very general statement holds except for one very specific case. Another example is the classification theorem for finite simple groups : a very general statement…
Hans-Peter Stricker
- 18,829
113
votes
5 answers
Getting Students to Not Fear Confusion
I'm a fifth year grad student, and I've taught several classes for freshmen and sophomores. This summer, as an "advanced" (whatever that means) grad student I got to teach an upper level class: Intro to Real Analysis. Since this was essentially…
Matt
- 7,698
113
votes
23 answers
Complete course of self-study
I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study mathematics. I request people to tell me what topics an…
user37450
113
votes
20 answers
Why is the volume of a sphere $\frac{4}{3}\pi r^3$?
I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$ , but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for…
Larry Wang
- 9,903
113
votes
6 answers
Why is the area under a curve the integral?
I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area under a curve for some reason when you break it up…
qwertymk
- 1,397
113
votes
6 answers
Why is a geometric progression called so?
Just curious about why geometric progression is called so. Is it related to geometry?
dark32
- 1,431
113
votes
4 answers
Continuous mapping on a compact metric space is uniformly continuous
I am struggling with this question: Prove or give a counterexample: If $f : X \to Y$ is a continuous mapping from a compact metric space $X$, then $f$ is uniformly continuous on $X$. Thanks for your help in advance.
the code
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112
votes
4 answers
Compute $\int_0^{\pi/4}\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)} x\exp(\frac{x^2-1}{x^2+1}) dx$
Compute the following integral \begin{equation} \int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, \exp\left[\frac{x^2-1}{x^2+1}\right]\, dx \end{equation} I was given two integral…
Anastasiya-Romanova 秀
- 17,399
112
votes
10 answers
How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?
Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to how to prove it is irreducible. Any…
MathTeacher
- 1,649
112
votes
3 answers
Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$
Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$. This question is just after the definition of differentiation and the theorem that if $f$ is finitely derivable at $c$, then $f$ is also continuous at $c$.…
Vikram
- 5,828
112
votes
2 answers
Does $\lfloor \sqrt{p} \rfloor$ generate all natural numbers?
Our algebra teacher usually gives us a paper of $20-30$ questions for our homework. But each week, he tells us to do all the questions which their number is on a specific form. For example, last week it was all the questions on the form of $3k+2$…
CODE
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