Example 1. A vector field is given in cylindrical coordinates $\mathbf{a}(M)=\mathbf{e}_{p}+\varphi \mathbf{e}_{\varphi}$. Find the vector lines of this field. We have
Solution. According to the problem's condition $a_{1}=1, a_{2}=\varphi_{1} a_{3}=0$. By formula (1)

$$
\frac{d \rho}{1}=\frac{\rho d \varphi}{\varphi}=\frac{d z}{0}
$$

Hence

$$
\left\{\begin{array}{l}
z=C_{1} \\
\rho=C_{2 \varphi}
\end{array}\right.
$$

this is a spiral of Archimedes, lying in planes parallel to the $x O y$ plane.

$2^{\circ}$. Gradient in orthogonal coordinates. Suppose we have a scalar field

$$
u=u\left(q_{1}, q_{2}, q_{3}\right)
$$

Then

$$
\operatorname{grad} u=\frac{1}{H_{1}} \frac{\partial u}{\partial q_{1}} \mathbf{e}_{1}+\frac{1}{H_{2}} \frac{\partial u}{\partial q_{2}} \mathbf{e}_{2}+\frac{1}{H_{3}} \frac{\partial u}{\partial q_{3}} \mathbf{e}_{3}
$$

In particular, in cylindrical coordinates ( $q_{1}=\rho_{1} q_{2}=\varphi, q_{3}=z$ ):

$$
\operatorname{grad} u=\frac{\partial u}{\partial \rho} \mathbf{e}_{\rho}+\frac{1}{\rho} \frac{\partial u}{\partial \varphi} \mathbf{e}_{\varphi}+\frac{\partial u}{\partial z} \mathbf{e}_{z}
$$

in spherical coordinates ( $q_{1}=r, q_{2}=\theta, q_{3}=\varphi$ ):

$$
\operatorname{grad} u=\frac{\partial u}{\partial r} \mathbf{e}_{r}+\frac{1}{r} \frac{\partial u}{\partial \theta} \mathbf{e}_{\theta}+\frac{1}{r \sin \theta} \frac{\partial u}{\partial \varphi} \mathbf{e}_{\omega-}
$$