A formula for the Jacobian of a flow - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:34:34Z http://mathoverflow.net/feeds/question/27423 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27423/a-formula-for-the-jacobian-of-a-flow A formula for the Jacobian of a flow Tom LaGatta 2010-06-08T03:57:38Z 2010-06-08T04:12:58Z <p>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 <a href="http://www.jstor.org/pss/1428307" rel="nofollow">the paper I'm reading</a>, 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.</p> <p>Could you please point me to a reference for this formula?</p> http://mathoverflow.net/questions/27423/a-formula-for-the-jacobian-of-a-flow/27426#27426 Answer by Andrey Rekalo for A formula for the Jacobian of a flow Andrey Rekalo 2010-06-08T04:12:58Z 2010-06-08T04:12:58Z <p>This is <a href="http://eom.springer.de/l/l059660.htm" rel="nofollow">the Liouville formula</a>. It is explained nicely in <em>Ordinary Differential Equations</em> by Arnold.</p>