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

In summary, the proof of the generalization begins with a proof of the original theorem, then shows that the generalization is true for a certain subset of the original domain, and finally shows that the generalization is also true for the entire original domain.
  • #1
Unconscious
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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?
 
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  • #2
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
Ok, the idea is clear. Thank you.
 
  • #4
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##.
 
Last edited:

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

What is the purpose of a change of variable in a multiple integral proof?

The purpose of a change of variable in a multiple integral proof is to simplify the integral and make it easier to solve. It involves replacing the original variables with new ones that make the integral easier to integrate.

What are the steps involved in a change of variable in a multiple integral proof?

The steps involved in a change of variable in a multiple integral proof are:

  1. Identifying the original variables and the limits of integration.
  2. Choosing new variables to replace the original ones.
  3. Calculating the Jacobian determinant of the new variables.
  4. Substituting the new variables and the Jacobian determinant into the original integral.
  5. Simplifying and solving the integral using the new variables.

Why is the Jacobian determinant important in a change of variable in a multiple integral proof?

The Jacobian determinant is important because it represents the change in volume under the transformation of variables. It is used to adjust the integral to account for the change in variables and ensure that the final result is accurate.

What are some common mistakes made when using a change of variable in a multiple integral proof?

Some common mistakes made when using a change of variable in a multiple integral proof include:

  • Choosing incorrect new variables.
  • Forgetting to include the Jacobian determinant in the substitution.
  • Using the wrong limits of integration for the new variables.
  • Not simplifying the integral correctly after the substitution.

How can I check if my change of variable in a multiple integral proof is correct?

You can check if your change of variable in a multiple integral proof is correct by:

  • Double-checking your calculations for the Jacobian determinant.
  • Substituting the new variables back into the original integral and seeing if it simplifies to the same result.
  • Integrating the original integral using the new variables and comparing the result to the original integral.

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