Paper 3, Section I, G

Consider the map $f: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ given by

$f(x, y, z)=(x+y+z, x y+y z+z x, x y z)$

Show that $f$ is differentiable everywhere and find its derivative.

Stating carefully any theorem that you quote, show that $f$ is locally invertible near a point $(x, y, z)$ unless $(x-y)(y-z)(z-x)=0$.

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