2.21. Calculate the circulation of the vector field:

a) $\vec{A}=x^{2} y^{2} \vec{i}+\vec{j}+z \vec{k}$ along the circle $x^{2}+y^{2}=a^{2}, z=0$;

b) $\dot{A}=(x-2 z) \dot{i}+(x+3 y+z) \dot{j}+(5 x+y) \vec{k}$ along the perimeter of the triangle $A B C$ with vertices $A(1,0,0), B(0,1,0), C(0,0,1)$.
Solution.

a) For $z=0$ we obtain the planar field $\vec{A}=x^{2} y^{2} \dot{i}+\ddot{j}$, where $P=x^{2} y^{2}, Q=1$. To compute the circulation of this field along the circle $x^{2}+y^{2}=a^{2}$, we can use Green's formula (2.62):

$$
\begin{gathered}
\oint_{x^{2}+y^{2}=a^{2}} \vec{A} \cdot d \vec{l}=\iint_{x^{2}+y^{2} \leqslant a^{2}}\left(Q_{x}-P_{y}\right) d x d y=-2 \iint_{x^{2}+y^{2} \leqslant a^{2}} x^{2} y d x d y= \\
\quad=-2 \int_{-a}^{+a} x^{2} d x \int_{-\sqrt{a^{2}-x^{2}}}^{+\sqrt{a^{2}-x^{2}}} y d y=-\left.2 \int_{-a}^{+a} x^{2} d x y^{2}\right|_{-\sqrt{a^{2}-x^{2}}} ^{\sqrt{a^{2}-x^{2}}}=0 .
\end{gathered}
$$

b) By Stokes' formula, the circulation of the vector field along the triangle $A B C$ is equal to the flux of $\operatorname{rot} \ddot{A}$ through the plane of the triangle $A B C$ with the outward normal (the normal forms an acute angle with the $O Z$ axis), i.e.,

$$
\oint_{A B C} \dot{A} \cdot d \dot{l}=\iint_{S_{A B C}}(\operatorname{rot} \dot{A} \cdot \dot{n}) d S
$$

We will calculate the necessary quantities entering the right-hand side of Stokes' formula:

$$
\begin{aligned}
& \operatorname{rot} \dot{A}=\left|\begin{array}{ccc}
i & j & \dot{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
x-2 z & x+3 y+z & 5 x+y
\end{array}\right|=-\dot{j} \cdot 7+\dot{k} ; \\
& S_{A B C}: x+y+z=1 ; \\
& \vec{n}=\frac{1}{\sqrt{3}}(1,1,1) ; \\
& d S=\sqrt{1+z_{x}^{2}+z_{y}^{2}} d x d y=\sqrt{3} d x d y .
\end{aligned}
$$

Thus, the sought circulation is:

$$
\oint_{A B C} \dot{A} \cdot \vec{l}=-6 \int_{\substack{x+y \leqslant 1 \\ x \geqslant 0 \\ y \geqslant 0}} d x d y=-6 \int_{0}^{1} d x \int_{0}^{1-x} d y=-3
$$

General problems related to scalar and vector fields.