Just as not all plane curves can be conveniently described as the graph of a function \(y=f(x)\) and are instead best described parametrically as \(\vec{r}(t) = x(t) \veci + y(t) \vecj\text{,}\) so surfaces are often decribed parametrically, but now with two parameters:
\begin{equation}
\vec{r}(u,v) = x(u,v) \veci + y(u,v) \vecj + z(u,v) \veck\tag{6.6.1}
\end{equation}
for \((u,v)\) in some domain \(D\) in \(\reals^2\text{.}\)
The set of all points \((x,y,z)\) in \(\reals^3\) given by this form is the parametric surface \(S\text{,}\) with the parametric equations
\begin{equation}
x = x(u,v), \; y=y(u,v), \; z = z(u,v)\tag{6.6.2}
\end{equation}
Tangent Planes.
To find the tangent plane to \(S\) at the point \(P(x_0,y_0,z_0)\) given by parameter values \((u_0,v_0)\text{,}\) one can first find two tangent directions,so that their cross product is a normal to the surface and to the tangent plane.
Natural choices are the tangent lines given by using the linearizations of \(x(u,v)\text{,}\) \(y(u,v)\) and \(z(u,v)\)at this point and varying one parameter while holding the other constant. Note that these are tangent lines to the grid curves.
Varying \(u\) while fixing \(v=v_0\) gives linearizations
\begin{align*}
x \amp= x(u_0,v_0) + \left[\frac{\partial x}{\partial u}(u_0,v_0)\right](u-u_0),\\
y \amp= y(u_0,v_0) + \left[\frac{\partial y}{\partial u}(u_0,v_0)\right](u-u_0),\\
z \amp= z(u_0,v_0) + \left[\frac{\partial z}{\partial u}(u_0,v_0)\right](u-u_0).
\end{align*}
This gives the tangent vector in the \(u\) direction
\begin{equation*}
\vec{r}_u = \frac{\partial x}{\partial u}(u_0,v_0) \veci + \frac{\partial y}{\partial u}(u_0,v_0) \vecj + \frac{\partial z}{\partial u}(u_0,v_0) \veck
\end{equation*}
Similarly, varying \(v\) with fixed \(u=u_0\) gives tangent vector
\begin{equation*}
\vec{r}_v = \frac{\partial x}{\partial v}(u_0,v_0) \veci + \frac{\partial y}{\partial v}(u_0,v_0) \vecj + \frac{\partial z}{\partial v}(u_0,v_0) \veck
\end{equation*}
If the cross product \(\vec{r}_u(u_0,v_0) \times \vec{r}_v(u_0,v_0)\) is non-zero then it is a normal vector to \(S\) at point \(P\) and the tangent plane there is
\begin{equation*}
(\vector{x,y,z} - \vector{x_0,y_0,z_0}) \cdot (\vec{r}_u(u_0,v_0) \times \vec{r}_v(u_0,v_0)) = \vec{0}
\end{equation*}
If the cross product \(\vec{r}_u \times \vec{r}_v\) is non-zero for all \((u,v)\) in \(D\text{,}\) this is caled a regular parameterization and the surface \(S\) is called smooth: loosely, it has no corners or creases, and there is a well-defined tangent plane everywhere on the surface.
This is analagous to the condition \(\vecr'(t) \neq \vec{0}\) in Section 3.2 for a curve to be smooth and with well-defined tangent vector.
