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 . Thenwhere 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

as .

**Step 2**. Applying the linear map represented by the matrix to get

Thus

which, by the triangle inequality, yields

Since the linear map is continuous, there is a constant such that

for any . Using this, we deduce that

Let be arbitrary but fixed. We also let sufficiently small in such a way that

for any . From this we obtain *the first fundamental estimate*

for any . In the next part of the proof, we aim to find a small ball centered at with smaller radius but compatible with . To this purpose, we crucially exploit the homemorphism of . The idea of such an argument comes from Mr. Tiến.

**Step 3**. Going back to the linear approximation for we get

which implies

Observe that

so long as with . Thus we have just obtained *the second fundamental estimate*

for all with .

**Step 4**. Since is open, is diffeomorphism (then homeomorphism), and is linear, the set

is open, which also contains the point . We denote this set by , namely, . Clearly, there is sufficiently small such that

From now on, we let

and observe that

It is now clear to see that

Indeed, let be arbitrary. There is some such that

Clearly, making use of the second fundamental estimate above gives

which gives the desired result.

**Step 5**. So far, we have already shown that

From this we deduce that

for some dimensional constant . It is now easy to verify that

First sending down to zero, then sending down to 0 we obtain the desired identity.

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