I've found the following example in a vector calculus book: the divergence of the vector field $\vec F(x,y,z) = x\vec i + y\vec j - z \vec k$ in spherical coordinates is
$$ \nabla \cdot \vec F(\rho,\phi,\theta) = -3\cos(2\phi). $$ I understood all the passages of the book and I've found the same result. But why the following argument does not holds true: in cartesian coordinates $\nabla \cdot \vec F(x,y,z)=1$ and then $\nabla \cdot \vec F(\rho,\phi,\theta)=1$? In my mind if $\nabla \vec G(x,y,z)=xy$ (for instance) then $\nabla \vec G(\rho,\phi,\theta) = \rho^2\sin\phi^2 \sin \theta \cos\theta$. Am I missing something?