Unclear step in "Change of variable in a multiple integral" proof

  • #1
Unconscious
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12
I'm studying the proof of this theorem (Zorich, Mathematical Analysis II, 1st ed., pag.136):

Screen Shot 2020-08-09 at 17.20.06.png


which as the main idea uses the fact that a diffeomorphism between two open sets can always be locally decomposed in a composition of elementary ones.
As a remark, an elementary diffeomorphism ##\varphi## is a diffeomorphism under which only a single variable is modified (in the following example, the last one):

##\left\{\begin{matrix}
x^1=\varphi^1(t^1,...,t^n)=t^1\\
x^2=\varphi^2(t^1,...,t^n)=t^2\\
...\\
x^{n-1}=\varphi^{n-1}(t^1,...,t^n)=t^{n-1}\\
x^{n}=\varphi^{n}(t^1,...,t^n)
\end{matrix}\right.##

The proposed proof (pag.142) starts in this way:

Screen Shot 2020-08-10 at 15.12.02.png


I can't understand the last sentence, because as an intuitively counterexample I'm imagining out this situation:

Screen Shot 2020-08-10 at 15.20.48.png


where the pink set is a set with a diameter less than ##\delta##, that intersect ##K_t##, but that is not contained in any neighborhoods of the finite family covering ##K_t##.
Am I missing something?
 

Answers and Replies

  • #2
pasmith
Homework Helper
2022 Award
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I think [itex]U(t_1), \dots, U(t_k)[/itex] is meant. The truth of that is a consequence of the triangle rule, and the consequence is that any sufficiently small neighbourhood is contained within a neighbourhood where [itex]\varphi[/itex] decomposes as claimed. (Compare the proof of the Heine-Cantor Theorem.)
 
  • #3
Unconscious
74
12
Ok, the idea is clear. Thank you.
 
  • #4
Unconscious
74
12
I post again in this thread, because I have another question on the same argument.
I'm studying this generalization (pag.145) of the previous theorem:

Screen Shot 2020-08-11 at 08.19.54.png


whose proof begins in this way:

Screen Shot 2020-08-11 at 11.59.33.png


where my doubt is on the highlighted relation.
I think that it would be ##x\in\partial D_x \cup S_x## instead ##x\in\partial D_x \setminus S_x##, no?
If I'm wrong, then I can't understand the successive relation ##\overline{V}_x\subset D_x\setminus S_x##, because I can't see why a point of ##\partial V_x## could not be inside ##S_x##.
 
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