Example 10. Find the direction of the greatest increase of the scalar field $u=x y+y z+x z$ at the point $M_{0}(1,1,1)$ and the magnitude of this greatest increase at this point.
Solution. The direction of the greatest change in the field is indicated by the vector grad $\chi(M)$. Let's find it:

$$
\operatorname{grad} u(M)=(y+z) \dagger+(x+c) j+(y+x) \mathbf{k}
$$

Thus, $\operatorname{grad} u(M)=2(\mathbf{I}+\mathbf{J}+\mathbf{k})$. This vector determines the direction of the greatest increase of the given field at the point $M_{0}(1, I, I)$. The magnitude of the greatest change in the field at this point is

$$
\max \frac{\partial u}{\theta l}=\left|\operatorname{grad} u\left(M_{\mathrm{n}}\right)\right|=2 \sqrt{3}
$$

## Problems for Independent Solution

69. Find the gradient of the scalar field $u=\ln \left(x^{2}+y^{2}+z^{2}\right)$ at the point $M_{0}(1,1,-1)$.
70. Find the gradient of the scalar field $u=x \mathrm{e}^{x^{2}+v^{2}+z^{2}}$ at the point $O(0,0,0)$. 71. Find the angle $\varphi$ between the gradients of the function $u=\operatorname{arctg} \frac{x}{y}$ at the points $M_{1}(1,1)$
and $M_{2}(-1,-1)$.
71. Find the angle $\varphi$ between the gradients of the function $u=(x+y) e^{\varepsilon+\gamma}$ at the points $M,(0,0)$ and $M_{2}(1, \mathbf{1})$
72. Find the angle $\varphi$ between the gradients of the function $u=\sqrt{x^{2}+y^{2}+x^{2}}$ and v $=\ln \left(x^{2}+\right.$ $y^{2}+z^{2}$ ) at the point $M_{2}(-1,-1)$.
73. Find the points where the gradient of the scalar field $u=\sin (x+y)$ equals $I+J$. 75. Find the points where the magnitude of the gradient of the scalar field

$$
u=\ln \sqrt{x^{2}+y^{2}+z^{2}}
$$

equals one.

76. Let $u=u(x, y, z)$ and $v=v(x, y, z)$ be differentiable at the point $M(x, y, z)$. Show that
a) $\operatorname{grad} \lambda u=\lambda \mathrm{grad} u, \lambda=$ const;
b) $\operatorname{grad}(u \pm v)=\operatorname{grad} u \pm \operatorname{grad} v$;
c) $\operatorname{grad}(u v)=v \operatorname{grad} u+u \operatorname{grad} v$;
d) $\operatorname{grad} \frac{u}{v}=\frac{v \mathrm{grad} u+u \operatorname{grad} v}{v^{2}}, v \neq 0$.
77. Show that

$$
\operatorname{grad} u(\varphi)=\frac{d u}{d \varphi} \operatorname{grad} \varphi
$$

where $\varphi=\varphi(x, y, z)$ is a differentiable function, $\iota=u(\varphi)$ has a derivative with respect to $\varphi$.

Find the gradients of the following scalar fields, if

$$
r=x|+y|+z k, \quad r=|r|=\sqrt{x^{2}+y^{2}+z^{2}}
$$

and $\mathbf{a}$ and $\mathbf{b}$ are constant vectors:
78. $u=\ln \mathbf{r}$.
79. $u=(a, r)$
80. $u=(\mathbf{a}, \mathbf{r}) \cdot(\mathbf{b}, \mathbf{r})$.
81. $u=|[\mathrm{a}, \mathrm{r}]|^{2}$

82. Show that $(\mathrm{grad} u(r), r)=u^{\prime}(r) \cdot r$.
83. Show that $[$ grad $u(r), r]=0$.
84. Let $w=f(u, v)$, where $u=u(x, y, x), v=v(x, y, z)$. Prove that

$$
\operatorname{grad} w=\frac{\partial f}{\partial u} \operatorname{grad} u+\frac{\partial f}{\partial v} \operatorname{grad} v,
$$

if $f, u, v$ are differentiable functions,

85. Let $G$ be a convex region in space (i.e., a region such that if two points $M$ and $N$ belong to the region $G$, then the entire segment $M N$ belongs to this region). Let a scalar field $u(M)$ be given in the region $G$, having a gradient at all points, continuous and bounded in $G$ :

$$
\mid \text { grad } u(M) \mid \leqslant A, \quad M \in G, \quad A=\text { const. }
$$

Prove that for any points $M$ and $N$ in the region $G$ the inequality holds

$$
|u(N)-u(M)| \leqslant A|\overrightarrow{M N}| .
$$

86. Find the derivative of the function $u=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}$ at the point $M(x, y, z)$ in the direction of the radius vector $\mathbf{r}$ of this point.
87. Find the derivative of the function $u=\frac{\mathbf{l}}{\mathbf{r}}$, where $\mathbf{r}=|\mathbf{r}|$ in the direction of the vector $\mathbf{l}=\cos$ al $+\cos \beta \mathbf{j}+\cos \gamma \mathbf{k}$. Under what condition is this derivative equal to zero?
88. Find the derivative of the function $u=\frac{1}{r}$, where $r=|r|$, in the direction of its gradient. 89. Find the derivative of the function $\boldsymbol{u}=y z e^{2}$ at the point $M_{0}(0,0,1)$ in the direction of its gradient.
89. Find the derivative of the scalar field $u=u(x, y, z)$ in the direction of the gradient of the scalar field $v=v(x, y, z)$. Under what condition is it equal to zero?
90. For the following scalar fields, find the direction and magnitude of the greatest change at the given points $M_{0}$ :

a) $u(M)=x^{2} y+y^{2} z+z^{2} x ; \quad M_{0}(1,0,0) ; \quad$ b) $u(M)=x y z ; \quad M_{0}(2,1,-1)$.

## CHAPTER

3

## Vector Field

## § 10. Vector Lines. Differential Equations of Vector Lines

Definition 1. If in each point $M$ of space or part of space a vector quantity $a=\mathrm{a}(\boldsymbol{M})$ is defined, then it is said that a vector field is given.

If a Cartesian coordinate system is defined in space, then the definition of a vector field $\mathbf{a} = \mathbf{a}(M)$ is equivalent to the definition of three scalar functions of the point $P(M), Q(M), R(M)$, such that

$$
\mathrm{a}(M)=P(x, y, z) \mathbf{i}+Q(x, y, z) \mathbf{j}+R(x, y, z) \mathbf{k}
$$

Definition 2. A vector line of a vector field $\mathbf{a}$ is a curve, at each point $M$ of which the vector $\mathbf{a}$ is directed along the tangent to this curve.

Let the vector field be defined by the vector

$$
\mathbf{a}=\boldsymbol{P} \mathbf{i}+Q \mathbf{j}+R \mathbf{k}
$$

where

$$
\boldsymbol{P}=\boldsymbol{P}(x, y, z), \quad Q=Q(x, y, z), \quad R=\boldsymbol{R}(x, y, z)
$$

are continuous functions of $x, y, z$, having bounded first-order partial derivatives.

Then the differential equations of the vector lines have the form

$$
\frac{d x}{P}=\frac{d y}{Q}=\frac{d z}{R}
$$

Integrating the system of two differential equations (1) gives a system of two constant equations

$$
\varphi_{1}(x, y, z)=C_{1}, \quad \varphi_{2}(x, y, z)=C_{2}
$$

which, considered together, define a two-parameter family of vector lines

$$
\left\{\begin{array}{l}
\varphi_{1}(x, y, z)=C_{11} \\
\varphi_{2}(x, y, z)=C_{2}
\end{array}\right.
$$

If in some region $G$ the conditions of the existence and uniqueness theorem for the system (1) are satisfied, then through each point $M_{0}\left(x_{0}, y_{0}, z_{0}\right) \in G$ there passes a unique vector line

$$
\left\{\begin{array}{l}
\varphi_{1}(x, y, z)=\varphi_{1}\left(x_{0}, y_{0}, z_{0}\right) \\
\varphi_{2}(x, y, z)=\varphi_{2}\left(x_{0}, y_{0}, z_{0}\right)
\end{array}\right.
$$