<s>[INST] Parallel surface For a regular surface $\mathbf{x} = \mathbf{x}(u,v)$ 
Define $\mathbf{y}(u,v) = \mathbf{x}(u,v) + t \mathbf{N} (u,v)$ where $\mathbf{N}$ is the unit normal of $\mathbf{x}$
How could I show the following?
$\mathbf{y}_u \times \mathbf{y}_v = (1-2Ht+Kt^2)\mathbf{x}_u \times \mathbf{x}_v$
where $H$ is the mean curvature of $\mathbf{x}$ and $K$ is its Gaussian curvature
 [/INST] 
Fundamental Forms
  $$
\begin{pmatrix} E & F \\ F & G \end{pmatrix}= 
\begin{pmatrix} \mathbf{x}_{u} \\ \mathbf{x}_{v} \end{pmatrix}
\begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix} $$
$$
 \begin{pmatrix} e & f \\ f & g \end{pmatrix}=
-\begin{pmatrix} \mathbf{N}_{u} \\ \mathbf{N}_{v} \end{pmatrix}
 \begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix} $$
Let $\begin{pmatrix} \mathbf{N}_{u} \\ \mathbf{N}_{v} \end{pmatrix}=
\mathbb{A} \begin{pmatrix} \mathbf{x}_{u} \\ \mathbf{x}_{v} \end{pmatrix}$ 
  where $\mathbb{A}=
\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$.
Now
  \begin{align*}
   \begin{pmatrix} \mathbf{N}_{u} \\ \mathbf{N}_{v} \end{pmatrix}
   \begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix} &=
   \mathbb{A} \begin{pmatrix} \mathbf{x}_{u} \\ \mathbf{x}_{v} \end{pmatrix}
   \begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix} \\
  -\begin{pmatrix} e & f \\ f & g \end{pmatrix} &=
   \mathbb{A} \begin{pmatrix} E & F \\ F & G \end{pmatrix} \\
   \mathbb{A} &=
  -\begin{pmatrix} e & f \\ f & g \end{pmatrix}
   \begin{pmatrix} E & F \\ F & G \end{pmatrix}^{-1}
\end{align*}
The principal curvatures $k_{1}, k_{2}$ are the eigenvalues of $-\mathbb{A}$.  That is
$$H=\frac{k_{1}+k_{2}}{2}=-\frac{1}{2} \operatorname{tr} \mathbb{A}$$
  $$K=k_{1} k_{2}=(-1)^{2} \det \mathbb{A}$$

\begin{align*}
  \mathbf{y}_{u} &= \mathbf{x}_{u}+t\mathbf{N}_{u} \\
  &= \mathbf{x}_{u}+t(a_{11} \mathbf{x}_{u}+a_{12} \mathbf{x}_{v}) \\
  \mathbf{y}_{v} &= \mathbf{x}_{v}+t\mathbf{N}_{v} \\
  &= \mathbf{x}_{v}+t(a_{21} \mathbf{x}_{u}+a_{22} \mathbf{x}_{v}) \\
  \mathbf{y}_{u} \times \mathbf{y}_{v} &=
  [1+t(a_{11}+a_{22})+t^{2}(a_{11}a_{22}-a_{12}a_{21})]
  \mathbf{x}_{u} \times \mathbf{x}_{v} \\
  &= (1-2Ht+Kt^{2}) \mathbf{x}_{u} \times \mathbf{x}_{v}
\end{align*}
</s>