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I need to find the Jacobi determinant for the unit sphere inversion in $\mathbb{R}^n$, i.e. the map given by $f(x) = \frac x {|x|^2}$ for $x\in \mathbb{R}^n$. The main problem is to figure out the determinant of the following matrix (when $\xi = f(x)$): $$\frac 1 {|\xi|^{4n}}\left[\begin{array}{cccc} |\xi|^2 - 2\xi_1^2 & -2\xi_1\xi_2 &\ldots& -2\xi_1\xi_n\\ -2\xi_2\xi_1 & |\xi|^2 - 2\xi_2^2 & \ldots& -2\xi_2\xi_n\\ \vdots & \vdots & \vdots & \vdots \\ -2\xi_n\xi_1 & \ldots & -2\xi_n\xi_{n-1} & |\xi|^2 - 2\xi_n^2 \end{array}\right]$$ Direct computation for low dimensions suggests that the result is $-\frac 1 {|\xi|^{2n}}$, but the recursive formula for n-th dimension seems terribly complicated, and involves all smaller dimensions. Is there a relatively simple way to solve this problem?

ajr
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1 Answers1

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First, the Jacobian is equal to $-1$ on the unit sphere. Indeed, consider an orthonormal basis at a point of the unit sphere where one of basis vectors is normal to the sphere. Under inversion, the normal flips its sign while the rest remain as they were: hence, the Jacobian matrix has eigenvalues $-1,1,1,\dots,1$.

Now take any $x_0$ with $|x_0|=r>0$. Observe that for all $x$,
$$ f(x) = r^{-1}f(r^{-1} x) $$ Taking the Jacobian on both sides yields $$ J_f(x) = r^{-n}J_f(r^{-1} x)r^{-n} = -r^{-2n} $$