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[Solved] Consider the equation z^3+z e^x+y+2=0. Show that the equation can be locally solved for z near the point (1,-1,-1), that is z=g(x, y) for some

Question: Consider the equation \(z^{3}+z e^{x+y}+2=0\).

  1. Show that the equation can be locally solved for \(z\) near the point \((1,-1,-1)\), that is \(z=g(x, y)\) for some continuously differentiable function \(g\) in an open set \(V \ni(1,-1,-1)\).
  2. Let
\[S=\left\{(x, y, z) \in \mathbb{R}^{3} \mid z^{3}+z e^{x+y}+2=0\right\}\]

be the set of solutions of the equation above. Is \(S\) a differentiable manifold (i.e a smooth surface)? What is the dimension of this manifold? Provide explanations.

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