I have a function $f:\Re^2\to\Re^2$ and would like to understand why $|Jf(x)|=\max_\theta|D_\theta f(x)|\cdot\min_\theta|D_\theta f(x)|$, that is, why the determinant of the Jacobian of $f$ at $x$ is equal to the product of min and max of the directional derivatives at $x$.

I think it must be a more general matrix fact, but I'm stuck at the moment.

I have a function $f: \Re^2 \to \Re^2$ and would like to understand why

$$|Jf(x)|=\max_\theta|D_\theta f(x)|\cdot\min_\theta|D_\theta f(x)|$$

that is, why the determinant of the Jacobian of $f$ at $x$ is equal to the product of the minimum and the maximum of the directional derivatives at $x$.

I think it must be a more general matrix fact, but I'm stuck at the moment.