Let $f$ be a function defined on the unit sphere $S^2$ parametrized by $(\theta, \phi)$ (as in spherical coordinates).
Is it true that $$\frac{\partial \Delta u}{\partial \theta}=\Delta\frac{\partial u}{\partial \theta}$$ and $$\frac{\partial \Delta u}{\partial \phi}=\Delta\frac{\partial u}{\partial \phi}?$$ In other words, does laplacian commute with partial derivatives with respect to $\theta$ and $\phi$?
Laplacian operator $\Delta$ in the coordinate can be tas: $$ \Delta u = \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial u}{\partial r} ) + \frac{1}{r^2 \sin^2\phi} \frac{\partial ^2 u}{\partial \theta ^2 } + \frac{1}{r^2 \sin\phi}\frac{\partial}{\partial \phi} ( \sin \phi \frac{\partial u}{\partial \phi}). $$ in case of $u(r,\phi,\theta)=u(\phi,\theta)$, and $r=1$ you get: $$ \Delta u = \frac{1}{ \sin^2\phi} \frac{\partial ^2 u}{\partial \theta ^2 } + \frac{1}{ \sin\phi}\frac{\partial}{\partial \phi} ( \sin \phi \frac{\partial u}{\partial \phi}). $$ as you can see coefficients of partial derivatives are functions of $\phi$ so they cannot commute with its partial derivatives respect to this variable. for example: $$ \frac{\partial}{\partial \phi} \bigg( \frac{1}{ \sin^2\phi} \frac{\partial ^2 u}{\partial \theta ^2 } \bigg) \neq \frac{1}{ \sin^2\phi} \frac{\partial ^2 }{\partial \theta ^2 } \bigg(\frac{\partial u}{\partial \phi} \bigg) $$ but they are not function of $\theta$. so with some assumption of continuity of $u$ for change the order of derivatives, you can change Laplacian with partial derivative respect to $\theta$.