[See Solution] Let f: R^2 \rightarrow R^2 be defined by (u, v)=f(x, y)=(x^3-3 x y^2, 3 x^2 y-y^3) Describe the set S of points in the xy -plane for which f has

Question: Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be defined by

\[(u, v)=f(x, y)=\left(x^{3}-3 x y^{2}, 3 x^{2} y-y^{3}\right)\]

Describe the set $S$ of points in the $xy$ -plane for which $f$ has a local inverse.
Suppose that $f\left(x_{0}, y_{0}\right)=\left(u_{0}, v_{0}\right)$ for some $\left(x_{0}, y_{0}\right) \in S$, and let $g$ be the inverse function for $f$ in a neighbourhood of $\left(u_{0}, v_{0}\right)$. Find the four first-order partial derivatives of $g$, that is find $\mathrm{D} g(u, v)$ for $(u, v)$ close to $\left(u_{0}, v_{0}\right)$

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[See Solution] Let f: R^2 \rightarrow R^2 be defined by (u, v)=f(x, y)=(x^3-3 x y^2, 3 x^2 y-y^3) Describe the set S of points in the xy -plane for which f has

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