Example 10. Calculate the line integral of the vector field given in cylindrical coordinates: $2=4 \rho \sin \varphi \mathbf{e}_{\rho}+z e^{p} \mathrm{e}_{,}+(\rho+\varphi) \mathbf{e}_{z}$, along the line $L:\left\{\varphi=\frac{\pi}{4}, z=0\right\}$ from point $O\left(0, \frac{\pi}{4}, 0\right)$ to point $A\left(1, \frac{\pi}{4}, 0\right)$
Solution. In this example,

$$
a_{\rho}=4 \rho \sin \varphi, \quad a_{p}=x e^{p}, \quad a_{z}=\rho+\varphi .
$$

By formula (13), the required line integral is

$$
\int_{L}(a, d r)=\int_{L} 4 \pi \sin \varphi d \rho+\rho z e^{\rho} d \varphi+(\rho+\varphi) d z
$$

On the line $\boldsymbol{L}$, we have:

$$
\varphi=\frac{\pi}{4}, d \varphi=0 ; \quad z=0, d z=0: \quad 0 \leqslant \rho \leqslant 1
$$

Therefore,

$$
\int_{L}(n, d r)=\int_{L} 2 \sqrt{2 \rho} d \rho=\sqrt{2} \int_{0}^{1} 2 \rho d \rho=\sqrt{2}
$$