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Let $U,V$ be two open sets in $R^n$ and $f:U\to V$ proper $C^{\infty}$ map (proper = preimage of compact set is compact). Then we have

$$\int f^{*}\omega=\deg(f)\int \omega,$$

for $\omega \in \Omega_c^{n}(V)$. How to prove that if $f$ is linear mapping i.e. $f(x)=Ax$ for nonsignular $n\times n$ matrix we have $\deg(A)=sign(\det (A))$?

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    I don't think that your property is true : we can show with Sard's theorem that the degree of a proper map is an integer (whereas your determinant could be a non-integer number).2012-06-16
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    Yes, I have corrected it. Sorry!2012-06-16

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