Example 3.37. Find the derivative of the function $u=\frac{1}{r^{2}}$, where $r^{2}=x^{2}+y^{2}+z^{2}$, in the direction of its fastest growth.
Solution. In this case, it is necessary to find the directional derivative along the gradient. This derivative, according to (3.27), is equal to $\frac{\partial u}{\partial \ell}=|\operatorname{grad} u|$ since $\cos (\angle \operatorname{grad} f, \vec{\ell})=1$. We determine the coordinates of the vector $\operatorname{grad} u$ at some arbitrary point $P(x, y, z)$:

$$
\frac{\partial u}{\partial x}=-\frac{2}{r^{3}} \frac{\partial r}{\partial x}=-\frac{4 x}{r^{3}} ; \frac{\partial u}{\partial y}=-\frac{4 y}{r^{3}} ; \frac{\partial u}{\partial z}=-\frac{4 z}{r^{3}}
$$

and $\operatorname{grad} u=-\frac{4}{r^{3}}(x \vec{i}+y \vec{j}+z \vec{k})$.

$$
\begin{aligned}
& \text { Then }|\operatorname{grad} u|=\sqrt{\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}+\left(\frac{\partial u}{\partial z}\right)^{2}}= \\
& =\frac{4}{r^{3}} \sqrt{x^{2}+y^{3}+z^{2}}=\frac{4}{r^{2}} .
\end{aligned}
$$

## 3.6. TANGENT PLANE AND NORMAL TO A SURFACE

Let the surface be given implicitly by the equation $F(x, y, z)=0$. Draw through the point $M(x, y, z)$ of this surface an arbitrary curve on the surface and construct a tangent to this curve at the indicated point (obviously, in the general case, there are an infinite number of such curves and tangents to them).

If all tangents at the point $M$ to the various curves passing through this point on the surface lie in one plane, then this plane is called the tangent plane to the surface at the point $M$, and the point $M$ itself is called the point of tangency.

A normal to the surface at the point $M$ is any non-zero vector parallel to the line passing through the point of tangency of the surface with the tangent plane constructed at the point $M$ and perpendicular to this plane. The specified line is called the normal line to the surface at the indicated point $M$.

If the surface is given implicitly by the equation $F(x, y, z)=0$ and has a tangent plane at some point $M\left(x_{0}, y_{0}, z_{0}\right)$, then the equation of this plane is

$$
\begin{aligned}
& F_{x}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)\left(x-x_{0}\right)+F_{y}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)\left(y-y_{0}\right)+ \\
& \quad+F_{z}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)\left(z-z_{0}\right)=0
\end{aligned}
$$

and the equation of the normal line is

$$
\frac{x-x_{0}}{F_{x}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)}=\frac{y-y_{0}}{F_{y}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)}=\frac{z-z_{0}}{F_{z}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)}
$$

In this case, the vector

$$
\vec{n}=\left\{\frac{\partial F(M)}{\partial x}, \frac{\partial F(M)}{\partial y}, \frac{\partial F(M)}{\partial z}\right\}
$$

can be taken as the normal at the specified point.

If the equation of the surface, having a tangent at the given point $M\left(x_{0}, y_{0}, z_{0}\right)$, is given explicitly as $z=f(x, y)$, then the equations of the tangent plane and the normal line at the point $M$ are respectively

$$
\begin{gathered}
z-z_{0}=f_{x}^{\prime}\left(x_{0}, y_{0}\right)\left(x-x_{0}\right)+f_{y}^{\prime}\left(x_{0}, y_{0}\right)\left(y-y_{0}\right) \\
\frac{x-x_{0}}{f_{x}^{\prime}\left(x_{0}, y_{0}\right)}=\frac{y-y_{0}}{f_{y}^{\prime}\left(x_{0}, y_{0}\right)}=\frac{z-z_{0}}{-1}
\end{gathered}
$$

It is easy to notice that the right-hand side of equation (3.32) represents, according to (3.5), the total differential of the function of two variables $z=f(x, y)$ at the point $P_{0}\left(x_{0}, y_{0}\right)$. Since $z-z_{0}$ represents the increment of the ordinate of the point of the tangent plane, it follows that the geometric meaning of the first-order differential of a function of two variables is: the first-order differential (or total differential) of the function of two variables $z=f(x, y)$ at the point $P_{0}\left(x_{0}, y_{0}\right)$ is the increment of the ordinate of the point of the tangent plane drawn to the surface at its point $M\left(x_{0}, y_{0}, z_{0}\right)$.