Suppose that the 2D vector field $$\boldsymbol{u} (x,y,z) = u_x (x,y,z) \hat{\boldsymbol{e}}_x + u_y (x,y,z) \hat{\boldsymbol{e}}_y$$ satisfies the property $$ \boldsymbol{\nabla}_\parallel \cdot \boldsymbol{u} = 0 \tag{1} $$ with $\boldsymbol{\nabla}_\parallel$ representing the tangential gradient, expressed in the system of Cartesian coordinates by $$ \boldsymbol{\nabla}_\parallel = \hat{\boldsymbol{e}}_x \, \frac{\partial}{\partial x} + \hat{\boldsymbol{e}}_y \, \frac{\partial}{\partial y} \, . $$ Accordingly, Eq. (1) is written in Cartesian coordinates as $$ \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0 \, . $$
Now I would like to express Eq. (1) in spherical coordinates in terms of $u_r$, $u_\theta$, and $u_\phi$, which represent the radial, polar, and azimuthal components of the vector field $\boldsymbol{u}$ in the system of spherical coordinates (using a physicist's notation.)
From the expression of the usual divergence in spherical coordinates we have ([Wikipedia][1]) $$ \boldsymbol{\nabla} \cdot \boldsymbol{u} = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 u_r \right) + \frac{1}{r\sin\theta} \frac{\partial}{\partial \theta} \left( u_\theta \sin\theta \right) +\frac{1}{r\sin\theta} \frac{\partial u_\phi}{\partial \phi} \, . $$
What I was able to get using an example vector field that $$ \boldsymbol{\nabla}_\parallel \cdot \boldsymbol{u} = \frac{u_r}{r} + \frac{1}{r\sin\theta} \frac{\partial}{\partial \theta} \left( u_\theta \sin\theta \right) +\frac{1}{r\sin\theta} \frac{\partial u_\phi}{\partial \phi} \, . $$
I have no idea why we get the first term $u_r/r$.
I would expect that the tangential divergence would only involve tangential contributions given that $$ \boldsymbol{\nabla}_\parallel f = \frac{1}{r} \frac{\partial f}{\partial \theta} \, \hat{\boldsymbol{e}}_\theta + \frac{1}{r \sin\theta} \frac{\partial f}{\partial \phi} \, \hat{\boldsymbol{e}}_\phi \, . $$
Any clarification is highly appreciated. Thank you!
In addition, how the tangential Laplace operator $$ \boldsymbol{\nabla}^2_\parallel \boldsymbol{u} = \left( \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} \right) u_x \, \hat{\boldsymbol{e}}_x + \left( \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} \right) u_y \, \hat{\boldsymbol{e}}_y $$ is defined in spherical coordinates?
[1]: https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates