Question 1: Suppose we have a unit 2-sphere, then the normal vector at point $(x,y,z)$ is vector $(x,y,z)$. So the divergence is $\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}=3$. But the normal vector can also be written as $(x,y,\sqrt {x^2+y^2})$, then the divergence is $\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial \sqrt {x^2+y^2}}{\partial z}=2$. What's the correct way to calculate the divergence of normal vector field?
Question 2: Suppose we have a unit 2-sphere, then the normal vector at point $(x,y,z)$ is vector $(x,y,z)$. So the divergence is $\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}=3$, but it's contradictory to the formula $\nabla \cdot \boldsymbol n = -2H$, where $H$ is the mean curvature $-1$. To calculate the divergence, we need to extend the normal vector field from the sphere to the ambient space, but the formula $\nabla \cdot \boldsymbol n = -2H$ doesn't put any constraints on the way of extension. So as the above, we get the result 3. Are there any constraints on the way of extension of the normal vector field?
Update: My question come from an integral:
$\int_{S^2} {\boldsymbol V \lrcorner d(\boldsymbol n \lrcorner (dx\wedge dy\wedge dz))}=\int_{S^2} {\boldsymbol V \lrcorner d((n_x,x_y,n_z) \lrcorner (dx\wedge dy\wedge dz))}=\int_{S^2} {\boldsymbol V \lrcorner d(n_x dy\wedge dz - n_y dx\wedge dz + n_z dx\wedge dy)}=\int_{S^2} {\boldsymbol V \lrcorner (\frac{\partial n_x}{\partial x}+\frac{\partial n_y}{\partial y}+\frac{\partial n_z}{\partial z})(dx\wedge dy\wedge dz)}=\int_{S^2} {(\frac{\partial n_x}{\partial x}+\frac{\partial n_y}{\partial y}+\frac{\partial n_z}{\partial z})(\boldsymbol V \cdot \boldsymbol n)\boldsymbol n \lrcorner(dx\wedge dy\wedge dz)}$. In the above integral, vector fields $\boldsymbol V, \boldsymbol n$ are only defined on the sphere surface, so we need to extend $\boldsymbol n$ to the ambient space to calculate the divergence. But it seems that no constraints are required on the way of extension.
So I wonder why must we extend it to an unit vector field?