Curvature With Parallel Surfaces

I have a two part question, one of which I have no clue of how to approach, the other has been giving me a difficult time for the last three days. The questions are as follow:

Le M be an arbitrairy orientable surface, with $\displaystyle U$ the unit normal vector. Define the following map: $\displaystyle F: M \mapsto \mathhbb{R}^3$ by $\displaystyle F(p)=p+\varepsilon U(p)$

**i)** Show that the canonical isomorphisms of $\displaystyle \mathhbb{R}^3$ make $\displaystyle U$ a unit normal on $\displaystyle \overline{M}=F(M)$ for which $\displaystyle \overline{S}(\overline{v})=S(v)$, where $\displaystyle v$ are vectors, and $\displaystyle S(v)$ indicates the shape operator.

I have no idea what's being asked here.

**ii)**Derive the following formulas of the Gaussian and Mean Curvature

$\displaystyle \overline{K}(F) = \frac{K}{J}$ where $\displaystyle J = (1-\varepsilon k_1)(1-\varepsilon k_2)$ and $\displaystyle k_i$ denote the principal vectors

Before I begin I managed to deduce from a previous problem that $\displaystyle (\overline{v})\times(\overline{w}) = J(p)v\times w$.

Now I would imagine that since $\displaystyle K=det(k_1,k_2)$ that I would simply have $\displaystyle F(k_1)\times F(k_2)$ wich would simply give $\displaystyle J(p)k_1 \times k_2$ which isn't the desired result.

So looking at both RHS and LHS I think I would get:

$\displaystyle F(k_1)\times F(k_2) = (k_1+\varepsilon k_1) \times (k_2+\varepsilon k_2)= k_1k_2(1-\varepsilon)^2$

then from the RHS I get :

$\displaystyle \frac{k_1k_2}{(1-\varepsilon k_1)(1-\varepsilon k_2)}$ which has lead nowhere.

I'm not looking for the answer, but any hints would be greatly appreciated.