Let $f : \Omega \rightarrow \mathbb{R}$ be a harmonic application of class $C^2$ such that $\partial_1 f$ and $\partial_2 f$ are of class $C^2$ on $\Omega$. Show that the applications $\partial_1 f$ and $\partial_2 f$ are also harmonic on $\Omega$.
Let $f : \Omega \rightarrow \mathbb{R}$ be a harmonic application of class $C^2$ such that $\partial_1 f$ and $\partial_2 f$ are of class $C^2$ on $\Omega$. Show that the applications $\partial_1 f$ and $\partial_2 f$ are also harmonic on $\Omega$.