Okay, let’s get to actually proving the change of variables formula for multiple integrals. To be explicit:
Let be a continuously differentiable function defined on an open region . Further, assume that is injective and that the Jacobian determinant is everywhere nonzero on . The inverse function theorem tells us that we can define a continuously differentiable inverse on all of the image .
Further, let be a Jordan measurable subset of , and let be defined and continuous on . Then we have the change of variables formula
We will proceed by induction on the dimension . For , this is exactly the one-dimensional change of variables formula, which we already know to be true. And so we’ll assume that it holds in all -dimensional cases, and prove that it then holds for -dimensional cases as well.
Since the Jacobian determinant is nonzero, we cannot have for all ; at least one of the components must have a nonzero partial derivative with respect to at any given point . Let’s say, to be definite, that . We will now (locally) factor into the composite of two functions and , which will each have their own useful properties. First, we will define
This is clearly continuously differentiable, and it’s even injective on some neighborhood of , by our assumption that . Further, the Jacobian determinant is exactly the partial derivative , and so the inverse function theorem tells us that in some neighborhood of we have a local inverse , with in some neighborhood of . We can now define
for , and define
Then for each in a small enough neighborhood of we have . The first function leaves all components of fixed except for , while the second function leaves fixed. Of course, if we used a different partial derivative , we could do the same thing, replacing instead with in , and so on.
Now if is a Jordan measurable compact subset of , then its inverse image will also be compact since is continuous. For every point in , we can find some neighborhood — which we can take to be an -dimensional interval — and a factorization of into two functions as above. As we move around, these neighborhoods form an open cover of . And since is compact, we can take an open subcover.
That is, we can cover by a finite collection of open intervals , and within each one we can write , where the function leaves all the components of fixed except for the last, while leaves that last one fixed. By subdividing these intervals, we can assume that they’re nonoverlapping. Of course, if we subdivided into open sets we’d miss the shared boundary between two subintervals. So we’ll include that boundary in each subinterval and still have nonoverlapping intervals .
Then we can define . Since is injective, these regions will also be nonoverlapping. And they’ll cover , just as covered . So we can define and write
Our proof will thus be complete if we can show that the change of variables formula holds for these regions, and we will pick up with this final step next time.