How to define Surface Laplacian on the sphere with radius 1 The simbol $\nabla_s f$ appears in a problem of my homework, and my professor thinks it means
$$\nabla_s f:= \nabla f - \hat{n}(\hat{n} \cdot \nabla f )$$
or 
$$ \nabla_s := (I - \hat{n}\hat{n}^T )\nabla $$
(the surface gradient of a function defined on a surface), where $f$ is a scalar field and $\hat{n}$ is the normal  surface vector (in this case the sphere of radius 1)
My question is, how can I define the ''surface Laplacian operator'' ($\nabla_s^2f$)  from the above definition? 
I need to find a way to calculate the following integral
$$ \int \int_{S^2} (u \nabla_s^2 v +\nabla_su\cdot\nabla_sv)dS$$
and I don't know how to calculate $\nabla_s^2 v $ for a given scalar field $v$ to (defined over the sphere)
Thanks for your help!

The surface gradient $\nabla_s f$ has 3 components (for a 2-dimensional hypersurface in $\mathbb{R}^3$). Let's call them
$$
   \nabla_s f = ( \underline{D}_1 f, \underline{D}_2 f, \underline{D}_3 f ).
$$
Then the surface Laplacian is simply:
$$
  \Delta_s f = \sum_{i=1}^3 \underline{D}_i \underline{D}_i f.
$$
