Example 11. Calculate the line integral of the vector field given in spherical coordinates: $2=e^{r}$ sin $\theta \mathbf{e}_{r}+3 \theta^{2}$ sin $\varphi \mathbf{e}_{\theta}+\tau \varphi \theta \mathbf{e}_{\varphi}$ along the line $L:\left\{r=1, \varphi=\frac{\pi}{2}, 0 \leqslant 0 \leqslant \frac{\pi}{2}\right\}$ in the direction from point $M_{0}\left(1,0, \frac{\pi}{2}\right)$ to point $M_{1}\left(1, \frac{\pi}{2}, \frac{\pi}{2}\right)$ (Fig. 40).
Solution. The line $L$ is a quarter circle with the center at the origin and radius $R=1$, located in the yOz plane. The coordinates of the given vector are

$$
a_{r}=e^{r} \sin \theta_{r} \quad a_{s}=3 \theta^{2} \sin \varphi_{,} \quad a_{p}=r \varphi \theta .
$$

According to formula (14), the line integral has the form

$$
\int_{L}(a, d r)=\int_{L} e^{r} \sin \theta d r+3 \theta^{2} r \sin \varphi d \theta+r^{2} \varphi \theta \sin \theta d \varphi
$$

Considering that on the line $L$ the conditions are satisfied:

$$
r=1, d r=0 ; \quad \varphi=\frac{\pi}{2}, d \varphi=0 ; \quad 0 \leqslant \theta \leqslant \frac{\pi}{2}
$$

we obtain

$$
\int_{L}(2 \cdot d r)=\int_{L} 3 \theta^{2} d \theta=\int_{0}^{\pi / 2} 3 \theta^{2} d \theta=\frac{\pi^{3}}{8}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_267d965536514099194eg-62.jpg?height=300&width=376&top_left_y=617&top_left_x=950)

Fig. 40

![](https://cdn.mathpix.com/cropped/2024_05_22_267d965536514099194eg-62.jpg?height=212&width=305&top_left_y=705&top_left_x=1342)

Fig. 41