Spivak's Calculus on Manifolds problem (I). Integration.

In summary, the conversation is about showing that if a set C is a subset of a rectangle A in n-dimensional space, then C is Jordan-measurable if and only if for every positive value of epsilon, there exists a partition P of A such that the difference between the sums of the volumes of the induced subrectangles containing points in C and those contained entirely within C is less than epsilon. The conversation also mentions that C is Jordan-measurable if its boundary has measure zero, and that the characteristic function of a set C is integrable in A. The attempt at a solution involves breaking down the sums of U and L, but the proof falls apart when trying to incorporate the other sum. The solution may involve considering the content of the
  • #1
ELESSAR TELKONT
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0

Homework Statement



If [tex]A\subset\mathbb{R}^{n}[/tex] is a rectangle show tath [tex]C\subset A[/tex] is Jordan-measurable iff [tex]\forall\epsilon>0,\, \exists P[/tex] (with [tex]P[/tex] a partition of [tex]A[/tex]) such that [tex]\sum_{S\in S_{1}}v(S)-\sum_{S\in S_{2}}v(S)<\epsilon[/tex] for [tex]S_{1}[/tex] the collection of all subrectangles [tex]S[/tex] induced by [tex]P[/tex] such that [tex]S\cap C\neq \emptyset[/tex] and [tex]S_{2}[/tex] the collection of all subrectangles [tex]S[/tex] induced by [tex]P[/tex] such that [tex]S\subset C[/tex].

Homework Equations



[tex]C[/tex] is Jordan-measurable iff [tex]\partial C[/tex] has measure zero.
The characteristic function of a set [tex]C[/tex] is the function [tex]\chi_{c}(x)=\left\{\begin{array}{cc}1 &x\in C\\ 0 &x\notin C\end{array}\right .[/tex]

The Attempt at a Solution



I have tried to proof directly (that means I suppose Jordan-measurable). Let [tex]\epsilon>0[/tex]. Then I use the fact that if [tex]C[/tex] is Jordan-measurable then [tex]\chi_{C}(x)[/tex] is integrable in [tex] A[/tex]. That means that [tex]U(\chi_{C},P)-L(\chi_{C},P)<\epsilon[/tex].

Now I could write [tex]U(\chi_{C},P)=\sum_{S\in S_{1}}v(S)+\sum_{S\in S_{2}}v(S)[/tex] because the supremum of the function at [tex]S_{1}[/tex] and [tex]S_{2}[/tex] is 1. In the same manner [tex]L(\chi_{C},P)=\sum_{S\in S_{2}}v(S)[/tex] because the infimum of the function at [tex]S_{1}[/tex] and [tex]S_{2}[/tex] is 0,1 respectively... Here my proof brokes since U-L as I have written U and L is [tex]U-L=\sum_{S\in S_{1}}v(S)<\epsilon[/tex]. I have no idea how to put into that the other sum. Any help that you can provide is precious.
 
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  • #2
Does the boundary of C have content zero? That may be the angle to start from (at least for the implication that assumes C is Jordan-measurable).
 

Related to Spivak's Calculus on Manifolds problem (I). Integration.

1. What is Spivak's Calculus on Manifolds problem (I)?

Spivak's Calculus on Manifolds problem (I) is a famous mathematical problem proposed by Michael Spivak in his book "Calculus on Manifolds". It involves integrating a function over a surface or manifold, with the goal of finding the area under the curve.

2. Why is Spivak's Calculus on Manifolds problem (I) significant?

This problem is significant because it requires a deep understanding of calculus and geometry, and it is often used as a benchmark for students studying these subjects. It also has practical applications in fields such as physics, engineering, and computer graphics.

3. What are some common techniques used to solve Spivak's Calculus on Manifolds problem (I)?

Some common techniques used to solve this problem include using parametric equations, converting the surface to a simpler form, and using multivariable calculus and differential forms. Other techniques may also be used depending on the specific problem.

4. Are there any tips for solving Spivak's Calculus on Manifolds problem (I)?

One tip is to break the problem down into smaller, more manageable parts. It can also be helpful to draw diagrams or visualize the problem in order to gain a better understanding. Additionally, practicing with similar problems and seeking help from a teacher or tutor can also improve problem-solving skills.

5. Can Spivak's Calculus on Manifolds problem (I) be solved using software or calculators?

Yes, there are software programs and calculators that can solve this problem. However, it is important to have a solid understanding of the underlying concepts and techniques in order to effectively use these tools. Relying solely on software or calculators may hinder the development of problem-solving skills.

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