Today, I went back to blogging and I decided to recreate my personal website to begin a new journey. My upcoming posts will be fascinating topics in Math, Physics and some interesting stories of daily life around us. Thank for following and supporting me !!
We will start the series of my posts with pleasant topics. In our earlier study of the integral in the one-dimensional case, we have already used an important formula for change of variable to compute some complicated integral. Such formula has the form
In this post, I would like to provide readers with a similar formula in the general case and its proof. The main result we will obtain is
Theorem 1. If is a diffeomorphism of a bounded open set onto a set , is a function integrable over and is supported in , then is integrable over . The following formula holds
We now divide the proof of this result into a few steps
Step 1: We now prove this result in the one-dimensional case. Firstly, we proceed with the proof for the case of are closed intervals in .
Indeed, let of through the mapping . We have from the uniform continuity of . Since Lagrange ‘s Theorem, .
We now write the Riemann sums for partitions with chosen points
Send , we obtain the formula (1).
Next, We point out that (1) holds for are arbitrary bounded open sets in . Let , It follows the boundedness and closedness of that is compact. Then, can be covered by a finite collection of closed intervals in such that two of which have no common interior points. Therefore,
Step 2: Now we prove (1) for the class of diffeomorphisms have the relatively “simple” form, that is element diffeomorphism. We recall the following definition.
Definition. The diffeomorphism is elementary if its coordinate representation has the form
We need to prove (1) hold for is elementary.
Indeed, assume without loss of generality that changes only the nth coordinate. Denote
Let with is a closed (n-1)-dimensional interval and is a closed interval of nth coordinate.
Respectively, we can choose an interval contains , taking account of the coincidence of first (n-1) coordinates of we can choose . By Fubini ‘s theorem,
Where is indicator function of the set . In this computation, we have already used the fact .
Step 3: Now we admit without proof the following result about the local resolution of the diffeomorphism. The proof will be provided in my later post.
Lemma 1. If is a diffeomorphism, then for any there is a neighborhood of the point in which the representation holds, where are elementary diffeomorphisms.
We need another result as a bridge to use the lemma 1, the result states as follows:
Lemma 2. If are diffeomorphisms for each of which formula (1) holds, then it holds also for the composition
Because the proof of Lemma 2 is quite simple, so it is left as an exercise for readers.
Complete the proof. In the final step, using results that we got above, we now provide a complete proof for the general case.
For each point of of the compact set , there exist a ball in which decomposes into a composition of elementary diffeomorphisms. Since the compactness of we can choose a finite covering of . Let . Then any set whose diameter is smaller than and its closure intersects must be contained in at least one of the neighborhoods .
Let is a collection of disjoint closed interval cover such that . Then intervals that have nonempty intersection with are contained in some and the integral over is equal to 0 iff it has an empty intersection with . Then,
Finally, Since , by Lemma 1, 2 and Step 2, we obtain that
Combining (2),(3), (4) , we get a complete proof for Theorem 1.
A straightforward corollary that we have often used is
Corollary . Let an injective, continuously differentiable such that the set has the measure zero. Then formula (1) holds for all integrable function .