Let $U : \mathbb R^d \to \mathbb R^d$ be a smooth vector field, and let $F\_t : \mathbb R \times \mathbb R^d \to \mathbb R^d$ be the corresponding smooth flow, defined by the differential equation $$\tfrac{d}{dt} F\_t(x) = U(F\_t(x)).$$ Let $f : \mathbb R^d \to \mathbb R$ be an integrable function, and consider the integral $$\int\_{\mathbb R^d} f( F\_t(x) ) ~dx.$$ To calculate this, I would make the change of variables $y = F\_t(x)$, making the ansatz that there exists some function $\rho(y)$ so that the integral equals $$\int\_{\mathbb R^d} f(y) \rho(y) ~dy.$$ From what I can gather from [the paper I'm reading](http://www.jstor.org/pss/1428307), it's common knowledge that this function $\rho$ has the form $$\rho(y) = \tfrac{1}{J\_t(F\_t^{-1}(y))} = \exp \left(-\int\_0^t (\operatorname{div} U)(F\_{s-t}(y) ) ~ds \right),$$ where of course $J\_t$ stands for the Jacobian of the flow.


Could you please point me to a reference for this formula?



A formula for the Jacobian of a flow
This is [the Liouville formula](http://eom.springer.de/l/l059660.htm). It is explained nicely in *Ordinary Differential Equations* by Arnold.


