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Jul
23
answered Why there is no sign of logic symbols in mathematical texts?
Jun
23
comment Electrostatic capacity of two spheres with changing radii
The problem might simplify in the appropriate choice of bispherical coordinates, where you the choose coordinate system so that both spheres are level sets. The potential is a solution of Laplace's equation in the space outside the conductors, which is separable in bispherical coordinates.
Jun
19
accepted Why are there only limits and colimits?
May
27
awarded  Popular Question
May
12
awarded  Great Answer
Apr
11
awarded  Yearling
Jan
4
asked Why are there only limits and colimits?
Nov
10
comment Determine whether there is linear independence or not
@dc3rd you can just pick another point. Plugging in $x=1$ would yield $ae+be^2=0$, and you could use that with $a+b=0$ to determine $a=b=0$ instead.
Nov
3
comment Is area under an integral limit exact or an approximation?
@user117913 There are no infinitesimal real numbers, so an "infinitely small" uncovered area is zero. There is no such thing in the real numbers as an infinitely small leftover. The integral is the limit, where the error is zero. Note that you can use other shapes than rectangles (trapezoids, for example) that better approximate the curve at intermediate steps, and you get the same answer in both cases. If there were some "infinitely small" error that came from using rectangles the answers should differ by a significant fraction of that error (and at any finite width, they do).
Oct
12
comment Do factorials really grow faster than exponential functions?
@CMCDragonkai $n^n$ grows faster. $n^n=n\times n\times n\times n\cdots$, while $n!=n\times(n-1)\times(n-2)\times(n-3)\cdots$. The factors in $n!$ are all smaller, so $n^n$ will be bigger. It's "allowed" to be bigger because the fact that the base is growing as well lets it stay ahead of the factorial.
Sep
30
awarded  Explainer
Jul
2
awarded  Curious
Jul
1
comment How can one prove that $\lim\limits_{\theta\to 0} \frac{\sin\theta}{\theta}=1$
@JohnJoy what's your definition of $\pi$ to start with? There are plenty that don't need to make any reference to trig functions whatsoever. They might assume facts that end up being logically equivalent to $\lim_{x\to 0} \sin x/x=1$, but I don't see how it's inevitably circular.
Jun
26
comment Do factorials really grow faster than exponential functions?
@Pacerier for a fixed $n$ that happens to be less than $a$, yes absolutely. But the point of the question (and answer) is in the behaviour of each function as $n$ grows arbitrarily large, and eventually a growing $n$ will always become bigger than any fixed $a$. So the question is meant to be read as "Why does $n!$ always eventually get (and stay) bigger than $a^n$ no matter what $a$ is?"
May
30
awarded  Nice Answer
May
29
comment How can Zeno's dichotomy paradox be disproved using mathematics?
@Hurkyl my understanding was that there isn't a significant difference to them. One says "he must always do something else first", the other says "he must always do something else before he passes". I know I wrote my sums in the backward order for this problem, but I only meant that for notational convenience.
May
29
answered How can Zeno's dichotomy paradox be disproved using mathematics?
May
16
comment How should I solve integrals of this type?
@JackM Differentiating $e^x\cos x$ or $e^x\sin x$ gives combinations of terms similar to your integrand. It stands that some combination of these might differentiate to give the integrand precisely; we then must figure out which combination.. You equate coefficients because you want to find the particular case where the (differentiated) functions are the same.
May
13
comment How to deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$
@Improve yes, with no extra effort. It's not a separate identical looking argument for the other roots or anything, you already have the entire result by that very statement you gave. Algebra can't tell the difference between the roots: it has no sense of 'where' they are, only the algebraic relationships they satisfy. So if it's not possible for one of them it can't be possible for any of them, because that would imply that algebraic relationships could somehow make a distinction between two quantities that satisfy all the same algebraic relationships, which is a contradictory statement.
May
12
answered How to deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$