Example 1. Calculate the circulation of the vector field $a=y i+z^{2} j-$ $z \mathbf{k}$ along the contour $L:\left\{\begin{array}{r}x^{2}+y^{2}=4, \\ z=3,\end{array}\right.$ 1) directly, 2) using Stokes' theorem.
Solution. 1) The contour $L-$ is a circle with a radius $=2$, lying in the plane $z=3$ (Fig. 32). We choose the orientation on it as indicated in the figure. Parametric equations of the line $\mathbf{L}$:

$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=3 \quad(0 \leqslant t<2 \pi)
\end{array}\right.
$$

BK 410

$$
d x=-2 \sin t d t, \quad d y=2 \cos t d t, \quad d z=0
$$

For the circulation of the vector field ฉ, we have

![](https://cdn.mathpix.com/cropped/2024_05_22_267d965536514099194eg-39.jpg?height=303&width=235&top_left_y=954&top_left_x=1432)

$$
\mu=\int_{0}^{2 \pi}\left[2 \sin t(-2 \sin t)+4 \cos ^{2} t 2 \cos t-3 \cdot 0\right] d t=-4 \pi
$$

2) For the calculation of the circulation using Stokes' theorem, we choose any surface $\Sigma$ stretched on the contour $L$. Naturally, we can take the disk whose boundary is the line $\boldsymbol{L}$. According to the chosen orientation of the contour, the normal $\mathbf{n}^{0}$ to the circle must be taken equal to $\mathbf{k}: \mathbf{n}^{\text {ї }}=\mathbf{k}$. We have

$$
\operatorname{rot} \mathrm{a}=\left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
y & x^{2} & -z
\end{array}\right|=(2 x-1) \mathbf{k}
$$

Therefore, by Stokes' theorem,

$$
\begin{aligned}
U=\iint_{\Sigma}\left(\operatorname{rot} \mathbf{a}, \mathbf{n}^{0}\right) d \sigma & =\iint_{\Sigma}(2 x-1) d \sigma= \\
& =\int_{0}^{2 \pi} d \varphi \int_{0}^{2}(2 \rho \cos \varphi-1) \rho d \rho=-\left.2 \pi \frac{\rho^{2}}{2}\right|_{0} ^{2}=-4 \pi
\end{aligned}
$$

## Problems for Independent Solution

194. Show that the flux of the rotor through an unclosed surface stretched on a given contour does not depend on the shape of the surface.

Find the circulation of vector fields along the specified contours directly and using Stokes' theorem.
195. $a=x \mathbf{i}+x \mathbf{j}+y \mathbf{k}$
L: $\left\{\begin{array}{l}x^{2}+y^{2}=4 \\ z=0 .\end{array}\right.$
196. $a=y \mathbf{i}-x \mathbf{j}+z \mathbf{k}$
L: $\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}=A_{1} \\ x^{2}+y^{2}=z^{2} \quad(z \geqslant 0) .\end{array}\right.$

197. $a=2 x z \mathbf{i}-y \mathbf{j}+z \mathbf{k}$ along the contour formed by the intersection of the plane $x+y+2 z=2$ with the coordinate planes.
198. $a=y \mathbf{i}-x \mathbf{j}+(x+y) \mathbf{k}$
L: $\left\{\begin{array}{l}z=x^{2}+y^{2} \\ z=1 .\end{array}\right.$
199. $a=z^{2} \mathbf{i}$;
$L:\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}=16 \\ x=0, y=0, z=0 .\end{array}\right.$
200. $a=z y^{2} \mathbf{i}+x z^{2} \mathbf{j}+x^{3} \mathbf{k}$
L: $\left\{\begin{array}{l}x=y^{2}+x^{2} \\ x=9 .\end{array}\right.$
201. $a=y^{2} \mathbf{i}+z^{2} \mathbf{j}$
L: $\left\{\begin{array}{l}x^{2}+y^{2}=9 \\ 3 y+4 z=5\end{array}\right.$
202. $a=y \mathbf{i}-x \mathbf{j}+z \mathbf{k}$;
L: $\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}=1, \\ x=z .\end{array}\right.$
203. Given the vector field of velocities $\mathbf{~ of ~ a ~ solid ~ body ~ rotating ~ with ~ a ~ constant ~ angular ~ velocity ~} \omega$ around the axis $\mathrm{Oz}$. Calculate the circulation of this field along the circle

$$
\left\{\begin{array}{l}
x=a \cos t \\
y=a \sin t \\
z=0 \quad(0 \leqslant t<2 \pi)
\end{array}\right.
$$

directly and using Stokes' theorem.

From Stokes' theorem, we obtain: the projection of the vector $\operatorname{rot} \mathbf{a}$ on any direction does not depend on the choice of the coordinate system and is equal to the surface density of the circulation of the vector field $\mathbf{a}$ along the contour of the section, perpendicular to this direction:

$$
\left.\operatorname{Ip} \operatorname{rot}_{\mathbf{n}}\right|_{M}=\left.\left(\operatorname{rot} \mathbf{a}, \mathbf{n}^{0}\right)\right|_{M}=\lim _{(\Sigma) \rightarrow M} \frac{\oint(\mathbf{a}, d \mathbf{r})}{S}
$$

Here $(\Sigma)$ is a flat area perpendicular to the vector $\mathbf{n}$, $S$ is the area of this flat area; $L$ is the contour of the section, oriented such that the traversal of the contour is seen from the tip of the vector $\mathbf{n}$ in the direction of the clock hands.

![](https://cdn.mathpix.com/cropped/2024_05_22_267d965536514099194eg-40.jpg?height=35&width=747&top_left_y=599&top_left_x=911)
in which the vector $\operatorname{rot} \mathbf{a}$ is considered, and the direction of the normal $\mathbf{n}$ to this section remains the same throughout.