Example 1. Form the equations of the tangent plane and normal to the surface $z=2 x^{2}+y^{2}$ at the point $P_{0}(1,-1,3)$.
Solution. We have: $z_{x}^{\prime}=4 x, z_{y}^{\prime}=2 y ; z_{x}^{\prime}(1,-1)=4, z_{y}^{\prime}(1,-1)=-2$. Therefore,

$$
\begin{aligned}
& (t): z=3+4(x-1)-2(y+1), \quad \text { or } \quad 4 x-8 y-z-3=0 \\
& (n): \frac{x-1}{4}=\frac{y+1}{-2}=\frac{z-3}{-1}
\end{aligned}
$$