Tangent map, gauss map, and shape operator

[badgers14]

Can anyone help me with this problem??
Let M be a surface in R^3 oriented by a unit normal vector field
U=g₁U₁+g₂U₂+g₃U₃
Then the Gauss map G:M$$\rightarrow$$$$\Sigma$$ of M sends each point p of M to the point (g₁(p),g₂(p),g₃(p)) of the unit sphere $$\Sigma$$.
Show that the shape operator of M is (minus) the tangent map of its Gauss Map: If S and G:M$$\rightarrow$$$$\Sigma$$ are both derived from U, then S(v) and -G*(v) are parallel for every tangent vector v to M.
Any help is appreciated. Thanks

 

[lavinia]

the normal map is the same as the Gauss map. Its derivative is the tangent map. the shape operator by definition is the negative of this - I think.