For a differentiable function \(f\) of three variables one can likewise define the directional derivative of in direction \(\vec{u}=\vector{u_1,u_2,u_3}\)
\begin{equation*}
D_{\vec{u}}f(x,y,z) = \lim_{h \to 0}\frac{f(x+u_1h,y+u_2h,z+u_3h)-f(x,y,z)}{h}
\end{equation*}
and the gradient
\begin{equation*}
\del f(x,y,z) = \vector{f_x(x,y,z),f_y(x,y,z),f_z(x,y,z)}.
\end{equation*}
Again
\begin{equation*}
D_{\vec{u}}f(x,y,z) = \vec{u} \cdot \del f(x,y,z),
\end{equation*}
and again the directional derivative is greatest in the direction of the gradient.
Theorem 4.6.4. The Implicit Function Theorem in 3D.
If at a point \(P(x_0,y_0,z_0)\) on a level set \(F(x,y,z)=k\) the gradient is non-zero then there is a differentiable parametric surface \(x=f(t,s), y=g(t,s), z=h(t,s)\) which passes through the point \((x_0,y_0,z_0)\) and lies in the level set \(F(x,y,z)=k\text{:}\) \(F(f(t,s), g(t,s), h(t,s))=k\text{.}\)
Further, the parameters can be chosen to be one of the pairs \((x,y)\text{,}\) \((x,z)\) or \((y,z)\text{,}\) which ensures that this part of the level set near the point \(P\) is a smooth surface approximated well by a tangent plane.
Tangent Planes to Level Surfaces.
For a function \(F\text{,}\) consider the level surface \(S\) given by \(F(x,y,z)=k\) through a point \(P(x_0,y_0,z_0)\text{,}\) consider and any curve \(\vec{r}(t) = \vector{x(t),y(t),z(t)}\) that lies in this level surface and passes through this point when \(t=t_0\text{.}\) The composition of \(F\) with these three component functions give the constant function \(F(x(t),y(t),z(t))=k\text{,}\) and by the Chain Rule, its zero derivative is also
\begin{equation*}
0 = \frac{\partial F}{\partial x}\frac{dx}{dt} + \frac{\partial F}{\partial y}\frac{dy}{dt} + \frac{\partial F}{\partial z}\frac{dz}{dt}
= \del F \cdot \vec{r}'(t)
\end{equation*}
In particular \(\del F(x_0,y_0,z_0) \cdot \vec{r}'(t_0) = 0\text{.}\)
The possible directions \(\vec{r}'(t_0)\) for a curve passing through this point and staying in the level curve are thus all orthogonal to \(\del F(x_0,y_0,z_0)\) and thus are directions in the plane:
\begin{align*}
\del \amp F(x_0,y_0,z_0) \cdot \vector{x-x_0,y-y_0,z-z_0}\\
\amp= F_x(x_0,y_0,z_0)(x-x_0) + F_y(x_0,y_0,z_0)(y-y_0) + F_z(x_0,y_0,z_0)(z-z_0) = 0
\end{align*}
Thus it is natural to call this plane the tangent plane to this level surface \(S\) at point \(P\) and to say that the direction of \(\del F(x_0,y_0,z_0)\) is normal to \(S\) at \(P\text{.}\)
Tangent Planes to Graphs.
An important special case is the tangent plane to the graph of a function of two variables, \(z=f(x,y)\text{.}\) This is given as a level surface by \(F(x,y,z)=f(x,y)-z=0\text{,}\) so \(F_z(x,y,z)=-1\text{,}\) leading to the equation
\begin{equation*}
f_x(x_0,y_0)(x-x_0)+f_x(x_0,y_0)(y-y_0) - (z-z_0)=0.
\end{equation*}
Using \(z_0=f(x_0,y_0)\text{,}\) this is
\begin{equation*}
z=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_x(x_0,y_0)(y-y_0),
\end{equation*}
which is the equation of the tangent plane to
\(z=f(x,y)\) at
\((x_0,y_0)\) seen in
Section 4.4.
