Can anyone give me a reference which explain the derivation of the partial differential operator expression for the laplacian on the euclidean n-dimensional space and on $S^n$ ?


One generally writes the laplacian on the n-dim euclidean space as a sum of a operator on the radial coordinate and $\frac{1}{r^2}$ times the laplacian on $S^n$. 


And very often the laplacian on $S^n$ is written through a recursion relation. 


I am looking for a reference which shows me the derivations of these. 



Looking for a reference for the laplacian operator 
The Laplacian can be defined on any Riemannian manifold as div grad. Here grad f for f a smooth function is the vector field dual to the 1-form df via the bilinear form of the metric. Div of a vector field X corresponds to taking the covariant derivative $\nabla X$, which is a (1,1) tensor, and taking the trace of that. In local coordinates one can give a formula using the symbols for the metric, which should yield what you are looking for. 


Another way to define div is to take the Lie derivative of the volume form: that is, $L\_X V = (div X) V$. The volume form depends on an orientation, which can be locally chosen. This way is actually probably easier for computing in local coordinates since you don't need to worry about a covariant derivative or Christoffel symbols.


For a reference, see e.g. Taylor's *Partial Differential Equations,* Vol. 1. In Folland's *Introduction to Partial Differential Equations,* there isn't much about Riemannian manifolds, but Folland does talk about how the Laplacian changes with respect to new coordinates.


