Jacobian determinant of diffeomorphisms measures the quotient of area of small balls

This post concerns a widely mentioned feature of the Jacobian determinant of diffeomorphisms whose proof is not easy to find. The precise statement of the result is as follows:

Geometric meaning of the Jacobian determinant: Let be open and be a diffeomorphism in . Fix a point . Then

where denotes the open ball in centered at with radius .

As a remark and to be more exact, we require to be a -diffeomorphism. Before proving the above result, it is worth noting that it is true for linear maps, whose proof is not hard. One way to realize this is to make use of the change of variable formula for multiple integrals. The proof presented here is inspired by the proof of Lemma 5.1.12 in this book.

We now proceed with the proof whose proof is divide into a few steps.

Step 1. First we use to denote a norm on . Clearly, because is a -diffeomorphism we can write

where the error vector-valued function enjoys the following properties