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Jan
28
revised What are the practical applications of the Taylor Series?
added 7 characters in body
Jan
28
comment Show that in a discrete metric space, every subset is both open and closed.
@Khallil that is also true, by the definition of 'closed'. But the complements are open as well here, since all sets are open. My point was to make the transition from "all sets are open" to "all sets are complements of open sets, and therefore closed as well".
Jan
25
awarded  Popular Question
Oct
25
comment Relation between $SO(n)$ and rotations
I guess I'm confused about the question. Rotations should be the operations which have a fixed center, do not flip orientation, and are rigid body transformations, which are precisely the properties used when defining $SO(n)$. Would you like a proof that such an operation can always be decomposed into a collection of 2D rotations around multiple planes (with one fixed axis in odd dimension)?
Sep
21
awarded  Popular Question
Sep
3
awarded  Enlightened
Sep
3
awarded  Nice Answer
Aug
26
awarded  Guru
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