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Given a smooth vector field $\mathbf{b}$ on $\mathbb{R}^n$, let $\mathbf{x}(s)=\mathbf{x}(s,x,t)$ solve the ODE $$\dot{\mathbf{x}}=\mathbf{b}(\mathbf{x}) (s\in\mathbb{R}), x(t)=x.$$

(a) Define the Jacobian $$J(s,x,t):=\det D_x\mathbf{x}(s,x,t)$$ and derive the Euler formula $$J_s=\operatorname{div} \mathbf{b}(\mathbf{x})J.$$

(b) Set $$u(x,t):=g(\mathbf{x}(0,x,t))J(0,x,t).$$ Show that $u(x,t)$ solves the equation $$u_t+\operatorname{div}(u\mathbf{b}(x))=0, \mbox{ in } \mathbb{R}^n\times\mathbb{R}_+,$$ with the initial value $u(x,0)=g(x)$.

This is an exercise in L. Evans' classic textbook "Partial differential equations (2nd Edition)", page 162-163. We have solved (a). Now we are focusing on (b). Evans gave a hint on (b): one should show $\frac{\partial}{\partial s}(u(\mathbf{x},s)J)=0$ firstly.

We have tried the characteristic method and some other methods, however, we are not able to work it out. Thanks for all of your help!

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