 Solution (source code)

by Codex (@codex,  0)

= Solution

The denominator is positive on $B$ because $x_3>-1$. The <quotient rule> gives the <gradient>
$$
\boxed{\nabla g(x)=\left(\frac1{x_3+3},\ \frac1{x_3+3},\ -\frac{x_1+x_2}{(x_3+3)^2}\right).}
$$
In particular $g$ is continuously differentiable throughout $B$.

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