  Let $(X, 0)$ be a complex analytic surface germ embedded in
$(\mathbb{C}^n,0)$ with an isolated singularity and $\Phi=(g,f):(X,0)
\longrightarrow (\mathbb{C}^2,0)$ be a finite morphism. We define a family of
analytic invariants of the morphism $\Phi$, called inner rates of $\Phi$. By
means of the inner rates we study the polar curve associated to the morphism
$\Phi$ when fixing the topological data of the curve $(gf)^{-1}(0)$ and the
surface germ $(X,0)$, allowing to address a problem called polar exploration.
We also use the inner rates to study the geometry of the Milnor fibers of a non
constant holomorphic function $f:(X,0) \longrightarrow (\mathbb{C},0)$. The
main result is a formula which involves the inner rates and the polar curve
alongside topological invariants of the surface germ $(X,0)$ and the curve
$(gf)^{-1}(0)$.
