Change of Variables in Multiple Integrals III
Today we finish up the proof of the change of variables formula for multiple integrals:
So far, we’ve shown that we can chop up into a collection of nonoverlapping regions and into their preimages . Further, within each we can factor , where fixes each component of except the last, and fixes that one. If we can show the formula holds for each such region, then it will hold for arbitrary (compact, Jordan measurable) .
From here we’ll just drop the subscripts to simplify our notation, since we’re just concerned with one of these regions at a time. We’ve got and its preimage . We’ll also define , so that . For each real we define
Then , since preserves the last component of the vector. We also define
The lowest and highest points in along the th coordinate direction. Now we can again define and set up the iterated integral
We can apply the inductive hypothesis to the inner integral using , which only involves the first coordinates anyway. If we also rename to , this gives
Which effectively integrates as runs over . But now we see that lies within the projection of , as we defined when we first discussed iterated integrals. We want to swap the order of integration here, so we have to rewrite the limits. To this end, we write , , and define
which runs over the part of above some fixed point in . Then we can reverse the order of integration to write
Now we can perform the one-dimensional change of variables on the inner integral and swap out the variables through to write
But now we recognize the product of the two Jacobian determinants as the Jacobian of the composition:
and so we can recombine the iterated integral into the -dimensional integral
Finally, since we can replace the composition. We can also replace by since there’s no chance anymore for any evaluation of to go outside . We’re left with
Essentially, what we’ve shown is that we can always arrange to first change variables (which we handle by induction) and then by the last variable. The overall scaling factor is the -dimensional scaling factor from the first transformation times the one-dimensional scaling factor from the second, and this works because of how the Jacobian works with compositions.
This establishes the change of variables formula for regions within which we can write as the composition of two functions, one of which fixes all but the last coordinate, and the other of which fixes that one. Since we established that we can always cut up a compact, Jordan measurable set into a finite number of such pieces, this establishes the change of variables formula in general.