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1500 questions
235
votes
10 answers
Exterior Derivative vs. Covariant Derivative vs. Lie Derivative
In differential geometry, there are several notions of differentiation, namely:
Exterior Derivative, $d$
Covariant Derivative/Connection, $\nabla$
Lie Derivative, $\mathcal{L}$.
I have listed them in order of appearance in my education/in…
Michael Albanese
- 99,526
234
votes
6 answers
When can you switch the order of limits?
Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \infty}\,\lim_{n\to \infty}{a_{nm}}$? Bonus points…
asmeurer
- 9,774
231
votes
13 answers
How can a piece of A4 paper be folded in exactly three equal parts?
This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on eye estimate where to fold. Then I turn the letter…
Nicky Hekster
- 49,281
230
votes
23 answers
How to check if a point is inside a rectangle?
There is a point $(x,y)$, and a rectangle $a(x_1,y_1),b(x_2,y_2),c(x_3,y_3),d(x_4,y_4)$, how can one check if the point inside the rectangle?
Freewind
- 2,525
229
votes
4 answers
How do I convince someone that $1+1=2$ may not necessarily be true?
Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical knowledge to convince each other.
And that is why,…
Aces12345
- 2,309
229
votes
11 answers
In simple English, what does it mean to be transcendental in math?
From Wikipedia, we have the following definitions:
A transcendental number is a real or complex number that is not algebraic
A transcendental function is an analytic function that does not satisfy a polynomial equation
However these definitions…
AlanSTACK
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- 6
- 27
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228
votes
4 answers
How many fours are needed to represent numbers up to $N$?
The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols.
For example, $165=\left(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}}}\right) \div .4$.
If we remove the restriction…
David Bevan
- 5,862
227
votes
8 answers
Proof that the trace of a matrix is the sum of its eigenvalues
I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.
JohnK
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- 4
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227
votes
1 answer
Does the open mapping theorem imply the Baire category theorem?
A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice.
On the other hand, the three classical consequences of the Baire category theorem in…
t.b.
- 78,116
227
votes
10 answers
Teaching myself differential topology and differential geometry
I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.
I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right…
Harddaysknight
- 2,271
226
votes
2 answers
Proving you *can't* make $2011$ out of $1,2,3,4$: nice twist on the usual
An undergraduate was telling me about a puzzle he'd found: the idea was to make $2011$ out of the numbers $1, 2, 3, 4, \ldots, n$ with the following rules/constraints: the numbers must stay in order, and you can only use $+$, $-$, $\times$, $/$, ^…
Kevin Buzzard
- 4,838
226
votes
2 answers
Is there a 0-1 law for the theory of groups?
For each first order sentence $\phi$ in the language of groups, define :
$$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } G}{\text{number of nonisomorphic groups of order} \le…
Dominik
- 14,396
225
votes
10 answers
What does $2^x$ really mean when $x$ is not an integer?
We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so please let me know what you think.
David G
- 4,277
224
votes
3 answers
When can a sum and integral be interchanged?
Let's say I have $\int_{0}^{\infty}\sum_{n = 0}^{\infty} f_{n}(x)\, dx$ with $f_{n}(x)$ being continuous functions. When can we interchange the integral and summation? Is $f_{n}(x) \geq 0$ for all $x$ and for all $n$ sufficient? How about when $\sum…
user192837
- 2,251
224
votes
11 answers
How do I sell out with abstract algebra?
My plan as an undergraduate was unequivocally to be a pure mathematician, working as an algebraist as a bigshot professor at a bigshot university. I'm graduating this month, and I didn't get into where I expected to get into. My letters were great…
Samuel Handwich
- 2,771