Tangent Plane to a regular surface I'd like some help with the following question:
Find the equation of the tangent plane in $(x_0,y_0,z_0)$ to a regular surface given by $f^{-1}(0)$, where $0$ is a regular value.
I tried to find local parametrization but It didn't work.
Thanks.
 A: If a surface is defined by a regular function $z=g(x,y)$, the equation of the tangent plane at $(x_{0},y_{0},z_{0})$ in terms of the  partial derivatives of $g$  is 
$$z=z_{0}+\left. \frac{\partial g}{\partial x}\right\vert
_{(x_{0},y_{0})}(x-x_{0})+\left. \frac{\partial g}{\partial y}\right\vert
_{(x_{0},y_{0})}(y-y_{0}).$$

If the surface is defined  implicitly by $f(x,y,z)=0$, then $z=g(x,y)=f^{-1}(0)$ (i.e. $f(x,y,g(x,y))\equiv 0$). Since
$$\frac{\partial g}{\partial x}=-\frac{\partial f}{\partial x}/\frac{\partial
f}{\partial z}$$
and
$$\frac{\partial g}{\partial y}=-\frac{\partial f}{\partial y}/\frac{\partial
f}{\partial z},$$
the equation of the tangent plane at $(x_{0},y_{0},z_{0})$ is given by
$$z=z_{0}-\left(\frac{\partial f}{\partial x}/\frac{\partial
f}{\partial z}\right)_{(x_{0},y_{0},z_{0})}(x-x_{0})-\left(\frac{\partial f}{\partial y}/\frac{\partial
f}{\partial z}\right)_{(x_{0},y_{0},z_{0})}(y-y_{0})$$
or
$$(x-x_{0})\left(\frac{\partial f}{\partial x}\right)_{(x_{0},y_{0},z_{0})}+(y-y_{0})\left(\frac{\partial f}{
\partial y}\right)_{(x_{0},y_{0},z_{0})}+(z-z_{0})\left(\frac{\partial f}{\partial z}\right)_{(x_{0},y_{0},z_{0})}=0.$$
In compact notation, $\mathbf{x}=\left( x,y,z\right) ,\mathbf{x}_{0}=\left(
x_0,y_0,z_0\right) $, we get
$$\left( \mathbf{x}-\mathbf{x}_{0}\right) \cdot \mathbf{\nabla }f(\mathbf{x}_{0})=0.$$
