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1 vote
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Hilberts hotel; mapping one type of infinity to another type

From comment: Some people in the second bus may have an infinite number of Bs (i.e. $1$s when translated) so for them you would not get a finite integer from your flipping
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Any set with Associativity, Left Identity, Left Inverse, is a Group. - Fraleigh p.49 4.38

I think, I found a pretty simple proof. Let's consider an equation $ e * a^{-1} * a $: Using the associativity we get $ e * (a * a^{-1}) = e * e = e = (e * a) * a^{-1} $ Since $ (e * a) * a^{-1} = e ...
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Intuitive proof of multivariable changing of variables formula (Jacobian) without using mapping and/or measure theory?

The answers here are good but I am tempted to add a part which I think is quite important and something which others haven't talked about from what I see: Namely, why are we allowed to use, in the ...
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1 vote

An interesting picture in differential geometry from the book *Visual Differential geometry and Forms by Needham* is not clear for me.

If you consider a $f:\mathbb R^3\to\mathbb R$ you have the concept of level surfaces: For each $p$ in the range of $f$, the level set at $p$ is $$S_p=f^{-1}(p)=\left\{v=(x,y,z)^{\top}\ :f(v)=p\right\}....
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Physical interpreation regarding heat equation

You're right in saying that this equation represents a conservation law. If you take the derivative out of the integrand, you end up with the integration of $kT,$ i.e., the total heat of the region. ...
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-1 votes

What should be the intuition when working with compactness?

I had a lot of difficulty understanding any of these answers because they have no pictures and I consider myself some what a visual learner, so this is an attempt at a Visual answer. Technical note: I'...
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The meaning and partial derivative of $f(x-y)$

You can imagine that partial derivative is a directional derivative. $F(x, y)$ is a function of two-dimensions, so you have a whole plain of directions. The derivative $\partial_x$ is the derivative ...
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The meaning and partial derivative of $f(x-y)$

I have a function $f(x−y)$, where $x,y∈\Bbb R$. Does taking a partial derivative with respect to $x$ or $y$ make sense here? Whenever $f$ is a derivable over $\Bbb R$, then $f(x-y)$ is a bivariate ...
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The meaning and partial derivative of $f(x-y)$

Answering your Initial Question: Does Partial Derivative make sense here ? Definition of Partial Derivative covers the case where $f$ is a function of 2 or more variables $x,y,...$ , hence $f(x-y)$ ...
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1 vote

Properties of Euler's totient function

Why (3) is true intuitively: For each $m_{i}$, $n_{j}$ respectively coprime and less than $m$, $n$, we can build one and only one $x_{ij}$ less than $m.n$ such that $x_{ij}$ has as remainder $m_{i}$ ...
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7 votes
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How to "feel" semidirect products

You are asking how to recognize a group as being a semidirect product, rather than how to build new groups using the semidirect product construction. Fact: a group $G$ is a semidirect product of two ...
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How to "feel" semidirect products

To elaborate on Cpc's answer, think about $G$ as a group presentation: that is, a set of words in some alphabet, subject to various algebraic simplification rules. $G$ is a direct product $G=H\times ...
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How to "feel" semidirect products

If $H,K\le G\,,HK=G$ and $H\cap K=e$, and only one of $H$ and $K$ is normal, then $G\cong H\rtimes K$. I think $H$ would be the normal one here. In general you need a homomorphism $\varphi: K\to\rm{...
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2 votes

Why is area under curve derived using $y\,\mathrm dx$ instead of $y\,\mathrm dl\;?$

The problem is that $\delta l \times y$ is not the area of the purple region in your illustration, that area is $\delta x \times y + \frac12\delta y\times\delta x$. To see this, let's use an extreme ...
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2 votes

Why is area under curve derived using $y\,\mathrm dx$ instead of $y\,\mathrm dl\;?$

Also known as a Riemann sum, the left hand rule version of a definite integral is $$\int_a^b f(x)dx=\lim_{N\to\infty}\frac{b-a}N\left(f(a)+ f\left(a+\frac{b-a}N\right)+ f\left(a+2\frac{b-a}N\right)+… ...
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4 votes
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Lcm of two numbers is a lot like the direct sum of two vector spaces.

Fix a field $F$, and consider all vector spaces over $F$ and linear transformations between them. Given two vector spaces $V$ and $W$, the direct sum $V\oplus W$ has the following properties: There ...
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1 vote

Moral justification for "sheaf=continuously variable set" and local injectivity

Perhaps the "justification" is to simply avoid duplicate copies of the same variable set, where one copy is the unramified version and the other is the ramified version. I think presheaves ...
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1 vote
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How can one understand almost-sure convergence in a physically meaningful way?

Yes, there is a more physically intuitive way of understanding almost-sure convergence! Let's first return to convergence in probability. Intuitively, this says: If you wait a very long time, then ...
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