Proof of Liouville's theorem for volume preservation

Question

I am trying to get my head around the following proof of Liouville's theorem (question is quite far down below):

Question: What I don't get my head around is the following line:

My guess is that it is some sort of coordinate change, since $\Phi_{t}(x)$ is a diffeomorphism: express $x$ as $x=\Phi_{s}(y)$, then we get $\Phi_{t}(x)=\Phi_{t}(\Phi_{s}(y))$. Now apply the chain rule

\begin{align} \frac{\partial\Phi_{t}(x)}{\partial x}(x)=\frac{\partial\Phi_{t}(x)}{\partial x}(\Phi_{s}(y))\frac{\partial\Phi_{s}(y)}{\partial y}(y) \end{align}

Then I would get something like

$$\det(\frac{\partial\Phi_{t}(x)}{\partial x}(x))=\det(\frac{\partial\Phi_{t}(x)}{\partial x}(\Phi_{s}(y))\det(\frac{\partial\Phi_{s}(y)}{\partial y}(y)),$$

which sort of looks the same as in the proof, but the $\Phi_{s}(x)$ in the denominator definitely confuses me.

I am also not too sure why differentiation and integration can be interchanged in the second bullet point. Is it really "obvious" that the requirements for this to be true are met?

Many thanks in advance!

Answer 1

Your interpretation is accurate; your source is using classical notation. A slightly tidier way of writing this is to simply consider the Jacobian cocycle $\delta(t,x)=|\det(D_x\Phi_t)|$. Calling $\delta$ a cocyle is to indicate that it satisfies the cocycle identity (by the chain rule):

$$\delta(t+s,x)= \delta(t,\Phi_s(x))\,\delta(s,x)$$

Now the main idea of the proof is that this cocycle is constant in the time variable $t$. For this the $t$ derivative is taken, but due to the cocycle identity the $t$ derivative at an anonymous time $t_0$ is related to the $t$ derivative at time $0$:

$$\lim_{h\to\infty}\frac{\delta(t_0+h,x)-\delta(t_0,x)}{h} = \lim_{h\to\infty}\frac{[\delta(h,\Phi_{t_0}(x))-1]\,\delta(t_0,x)}{h} = \lim_{h\to\infty}\frac{[\delta(h,\Phi_{t_0}(x))-\delta(0,\Phi_{t_0}(x))]\,\delta(t_0,x)}{h}. $$

Thus we have

$$\left.\frac{d}{dt}\right|_{t=t_0}\delta(t,x) = \left[\left.\frac{d}{dt}\right|_{t=0}\delta(t,\Phi_{t_0}(x))\right] \delta(t_0,x). $$

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Regarding the second question, $D$ is implicitly assumed to be a set over which differentiation under the integral sign is valid (e.g. if $D$ has finite volume for small enough $t$, $D(t)$ will also have finite volume and one can apply the dominated convergence theorem).