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Integrability of modified diagonalizable Jacobian

I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as $$ J_f(x)=S(x)\Lambda(x) {S(x)}^{-1}, $$ where the ...
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