Example. Find the circulation of the vector field

$$
\vec{a}=\frac{y}{3} \vec{i} + 3 - 3 x \vec{j} + x \vec{k}
$$

along the closed contour $\Gamma$

$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=1-2 \cos t-2 \sin t
\end{array} t \in[0,2 \pi]\right.
$$
Solution.

1. By definition, the circulation of a vector field is equal to the second kind of curvilinear integral along the curve $\Gamma$:

$$
A=\oint_{\Gamma}(\vec{a}, d \vec{r})=\oint_{\Gamma} \frac{y}{3} d x-3 x d y+x d z
$$

2. We compute the curvilinear integral by reducing it to a definite integral:

$$
A=\oint_{\Gamma} \frac{y}{3} d x-3 x d y+x d z=
$$

$$
=\int_{0}^{2 \pi}\left(-\frac{4}{3} \sin ^{2} t-16 \cos ^{2} t+4 \sin t \cos t\right) d t=-\frac{52 \pi}{3}
$$

Answer. $A=-52 \pi / 3$.

Conditions of the Problems. Find the circulation of the vector field $\vec{a}$ along the given closed contour $(t \in[0,2 \pi])$.

1. $\vec{a}=y \vec{i}-z^{2} \vec{j}+x^{2} y \vec{k}, \quad x=2 \cos t, y=\sin t, z=1$.
2. $\vec{a}=-x^{2} y \vec{i}+4 \vec{j}+x^{2} z \vec{k}, \quad x=2 \cos t, y=2 \sin t, z=4$.
3. $\vec{a}=x^{3} \vec{i}-z^{2} \vec{j}+y \vec{k}, \quad x=\cos t, y=2 \sin t, z=2 \cos t$.
4. $\vec{a}=-2 z \vec{i}-x^{2} \vec{j}+z \vec{k}, x=2 \cos t, y=2 \sin t, z=\sin t$.
5. $\vec{a}=y^{2} \vec{i}-z \vec{j}+y \vec{k}, \quad x=5 \cos t, y=2 \sin t, z=2(1-\cos t)$.
6. $\vec{a}=x^{3} \vec{i}-y^{2} \vec{j}+y \vec{k}, \quad x=\cos t, y=3 \sin t, z=\cos t-\sin t$.
7. $\vec{a}=y \vec{i}+x \vec{j}+y^{4} \vec{k}, x=2 \cos t, y=\sin t, z=5 \sin t$.
8. $\vec{a}=z \vec{i}-x \vec{j}+y z \vec{k}, \quad x=\cos t, y=6 \sin t, z=3$.
9. $\vec{a}=-x^{2} \vec{i}+5 \vec{j}+y \vec{k}, \quad x=\cos t, y=5 \sin t, z=3 \cos t$.
10. $\vec{a}=4 y \vec{i}+x \vec{j}+y \vec{k}, \quad x=\cos t, y=\sin t, z=2-\cos t-\sin t$.

Answers. 1. $-2 \pi$. 2. $18 \pi$. 3. $-4 \pi$. 4. $4 \pi$. 5. $-8 \pi$. 6. $-3 \pi$. 7. $\pi$. 8. $-6 \pi$. 9. $-15 \pi$. $10 .-8 \pi$.

## 14.8. Calculation of Circulation Using Stokes' Formula

Problem Statement. Find the modulus of the circulation of the vector field

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

along the closed contour

$$
\Gamma=\left\{\begin{array}{ll}
(x, y, z): \begin{array}{l}
F_{1}(x, y, z)=0 \\
F_{2}(x, y, z)=0
\end{array}
\end{array}\right\}
$$

Plan of Solution. The circulation $A$ of the vector field $\vec{a}=\{P, Q, R\}$ along a closed piecewise-smooth contour $\Gamma$, located in a region $G$ where the functions $P, Q, R$ have continuous partial derivatives, is equal to the flux of the curl of this vector field through any piecewise-smooth surface $\Sigma$ stretched on the contour $\Gamma$ and located in the region $G$:

![](https://cdn.mathpix.com/cropped/2024_05_22_5878ce417e2ca061000bg-363.jpg?height=39&width=992&top_left_y=574&top_left_x=69)

$$
A=\iint_{\Sigma}\left(\operatorname{rot} \vec{a}, \overrightarrow{n_{0}}\right) d \sigma
$$

It is assumed that the orientation of the unit normals

$$
\overrightarrow{n_{0}}=\{\cos \alpha, \cos \beta, \cos \gamma\}
$$

to the surface $\Sigma$ is consistent with the orientation of the contour $\Gamma$ such that from the end of each normal, the traversal of the contour in the chosen direction appears as a counterclockwise traversal.

1. There are two opposite directions for traversing $\Gamma$. The circulation for these traversals differs only in sign. Since only the modulus of the circulation is required, we choose the traversal direction arbitrarily and write the modulus of the result in the answer.
2. Choose a surface $\Sigma$ stretched on the contour $\Gamma$.
3. Determine the normals $\overrightarrow{n_{0}}$ to the surface $\Sigma$.
4. Find $\operatorname{rot} \vec{a}$:

$$
\operatorname{rot} \vec{a}=\left|\begin{array}{ccc}
\vec{i} & \vec{j} & \vec{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
P & Q & R
\end{array}\right|
$$

5. Compute the scalar product $\left(\operatorname{rot} \vec{a}, \overrightarrow{n_{0}}\right)$.
6. Apply Stokes' formula (1):

$$
A=\iint_{\Sigma}\left(\operatorname{rot} \vec{a}, \overrightarrow{n_{0}}\right) d \sigma
$$

7. Compute the surface integral.

Write the answer, remembering to include the modulus.