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I get confused in the following problem. Actually I don't know how to derive the derivative of the Jacobian. Could anybody help me?

Given a smooth vector field $\vec{b}$ on $R^{n}$, let $\vec{X}(s)=\vec{X}(s,x,t)$solves the ODE:

$\dot{\vec{X}}=\vec{b}(\vec{X})(s \in R)$,

$\vec{X}(t)=\vec{X}$

Define the Jacobian $J(s,x,t)=det D_{x}\vec{X}(s,x,t)$, derive Euler's formula:

$J_{s}=(div\vec{b}(\vec{X}))J$

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    I am confused by the statement. I understand $t$ is the independent variable of the ODE. What are $s$ and $x$?2011-01-09

1 Answers 1

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HINT

The derivative of the determinant can be evaluated using the adjugate. Then it remains to compute

$$ \frac{d}{ds} D_x X(s,x,t) $$

for which you need to use the ODE.

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    I know that $\frac{d}{d\alpha}det(A)=tr(adj(A)\frac{d A}{d\alpha})$, and $\frac{d}{ds}D_{x}(\vec{X})=(\frac{\partial b_{i}(\vec{X})}{\partial x_{j}})_{i,j}$, but then it turns out that $tr(\frac{d}{ds}D_{x}(\vec{X}))=div(\vec{b}(\vec{X}))$, which is not I want, since the $adj(D_{x}(\vec{X}))$ term is missing.2010-11-10