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Jun
10
comment surface area of the graph of a convex function
Of course my definition for "area" is the $n-1$ dimensional Hausdorff measure. I suppose the last part of your answer doesn't restrict itself to 3 dimension. The argument works for any dimension with no modification. Thanks for this viewpoint. I was tempted to ask about the proof when $f$ is not differentiable at all.
May
13
comment Min-Max Principle and Harnack's inequality
Thanks for this very well explained answer.
May
10
comment Min-Max Principle and Harnack's inequality
@GiuseppeNegro just trying to see if somebody might have an idea. I am not "requiring" people to know or answer. That's the purpose of this website, right? People raise questions and people discuss.
May
10
comment Min-Max Principle and Harnack's inequality
@GiuseppeNegro It was merely a sentence saying that. I can't find it again... sorry. But as for my doubt, there is at least no easy way to prove maximum principle with harnack, right?
Apr
6
comment proof on Poincare's inequality.
which notation do you need me to explain?
Mar
29
comment Limiting argument when proving inequality in Sobolev space
this makes a hell lot more sense!! Thanks!
Mar
29
comment Limiting argument when proving inequality in Sobolev space
So I should interpret this inequality as: For any $f$ in $W^{1,1}$, there is a $\tilde{f}$, s.t. $\tilde{f}=f$ in the distributional sense with $\tilde{f}\in L^{\infty}$ and that inequality holds. Am I right?
Mar
29
comment Limiting argument when proving inequality in Sobolev space
first thank you for pointing out the Cauchy sequence part. I was stupid to not have seen this. But why does $f_n$ necessarily converge to this same $f$ in $L^{\infty}$?
Mar
21
comment What does it mean for a distribution to be in $L_2$?
I am trying to get a little intuition here. So based on what you said, a lot of distributions are not in $L_2$ since for a general distribution, you are not even lucky enough to get that kind of representation. Am I right?
Feb
10
comment What is the characteristics for the wave equation with space dimension more than 1?
What do you mean by there are no characteristics in higher spatial dimensions? I thought we need to just find a function $\phi(t,x,y)$, s.t. $\phi_t^2=\phi_x^2+\phi_y^2$. And the set$\{\phi=const\}$ would be the family of characteristic surfaces. When I tried to solve the equation, I got planes. But my intuition tells me that it should be something like the cone you mentioned. I don't know where I went wrong.
Apr
8
comment Approximating measurable function by continuous ones
@GiuseppeNegro I did check on that one. But isn't regularity of the measure needed there?
Apr
8
comment Approximating measurable function by continuous ones
@DavideGiraudo Let's say $X$ is $\mathbb{R}^n$ with the usual topology. And sorry that I made a mistake in my original problem. I should ask for almost everywhere convergence.
Apr
8
comment Steiner symmetrization preserves area?
This is just by the construction of Steiner symmetric action and Fubini's theorem( where you can get: in order to get area, you integrate length of chords with respect to proper measure)
Feb
12
comment Topology on set of maps between manifolds
@Sigur thanks, man
Feb
10
comment Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book)
I've actually been looking at this proof at a more careful level. And I realized Tubular Neighborhood and Retraction are exactly the tools that Lee used to solve the question I have. I guess I will choose @Potato as the best answer if nobody gives a better answer in the following couple of days! Thanks man, @Potato!
Feb
9
comment Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book)
I saw this proof before. But since I am not aware of Whitney approximation theorem or the smooth retraction mentioned in the proof, I skipped it. Bott, Tu's book's proof requires less background knowledge. But as I explained, I found I can't get around that point. I still wish that someone might be able to tell me how it is possible.
Nov
13
comment How to prove this formula for Lie Derivative for differential forms
@FlybyNight I believe $\phi_t$ in my notation is the $f$ in your notation. And $\phi_t^* w$, viewed as a whole thing, is exactly the $w^*$ in your notation. $\phi_t^*$ is the mapping that takes $ w$ to $w^*$. But never mind, I know how to prove this thing now. It's only a simple computation once you know what all those mappings really are. Thanks anyway.
Nov
13
comment How to prove this formula for Lie Derivative for differential forms
@FlybyNight huh? $w$ here is a differential form.
Sep
26
comment How do I see that the tangent bundle of torus is trivial
Thanks, @Neal, for providing so many ways of seeing it
Sep
26
comment How do I see that the tangent bundle of torus is trivial
@RyanBudney Thanks! I guess that's one way of doing it.