Example. Find the work of the force

\[
\vec{F}=(x-y) \vec{i}+\vec{j}
\]

when moving along the curve \( L \)

\[
x^{2}+y^{2}=4 \quad(y \geq 0)
\]

from point \( M(2,0) \) to point \( N(-2,0) \).
Solution.

1. The work $A$ of a force field is equal to the line integral of the second kind along the curve $L$:

$$
A=\int_{L}(\vec{F}, d \vec{r})=\int_{L}(x-y) d x+d y
$$

2. We compute the line integral. For this:

a) since $L$ is the upper semicircle, its parametric equations are written as

$$
\left\{\begin{array}{l}
x=2 \cos t, \\
y=2 \sin t,
\end{array} \quad 0 \leq t \leq \pi\right.
$$

We compute $d x=-2 \sin t d t$ and $d y=2 \cos t d t$;

b) we transform the line integral into a definite integral:

$$
A=\int_{L}(x-y) d x+d y=\int_{0}^{\pi}[(2 \cos t-2 \sin t)(-2 \sin t)+2 \cos t] d t=2 \pi
$$

Answer. $A=2 \pi$ units of work.

Conditions of the Problem. Find the work of the force $\vec{F}$ when moving along the given curve from point $M$ to point $N$.

1. $\quad \vec{F}=x \vec{i}+y \vec{j}, \quad x^{2}+y^{2}=4(y \geq 0), \quad M(2,0), N(-2,0)$.
2. $\vec{F}=x^{3} \vec{i}, \quad x^{2}+y^{2}=4 \quad(y \geq 0, x \geq 0), \quad M(2,0), N(0,2)$.
3. $\vec{F}=\left(x^{2}+y\right) \vec{i}+\left(x^{2}-y\right) \vec{j}, \quad y=x^{2}, \quad M(-1,1), N(1,1)$.
4. $\vec{F}=2 x y \vec{i}+x^{2} \vec{j}, \quad y=2 x^{3}, \quad M(1,2), N(2,16)$.
5. $\vec{F}=y \vec{i}+2 y \vec{j}, \quad x^{2}+y^{2}=1(y \geq 0), \quad M(1,0), \quad N(-1,0)$.
6. $\vec{F}=x \vec{i}-y \vec{j}, \quad x^{2}+y^{2}=2(y \geq 0), \quad M(\sqrt{2}, 0), \quad N(-\sqrt{2}, 0)$.
7. $\vec{F}=2 y \vec{j}, \quad x^{2}+y^{2}=1(y \geq 0, x \geq 0), \quad M(1,0), \quad N(0,1)$.
8. $\vec{F}=x^{2} y \vec{i}, \quad x^{2}+y^{2}=4(y \geq 0, x \geq 0), \quad M(2,0), \quad N(0,2)$.
9. $\vec{F}=\left(y-x^{2}\right) \vec{i}+x \vec{j}, \quad y=2 x^{2}, \quad M(0,0), \quad N(1,2)$.
10. $\vec{F}=y \vec{i}-x \vec{j}, \quad y=x^{3}, \quad M(0,0), \quad N(2,8)$.

Answers. 1. 0 units of work. 2. -4 units of work. 3. 4/3 units of work. 4. 62 units of work. 5. $-\pi$ units of work. 6. 0 units of work. 7. 1 unit of work. 8. $-\pi$ units of work. 9. 5/3 units of work. 10. -8 units of work.

## 14.7. Circulation of a Vector Field

Problem Statement. Find 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 $\Gamma$

$$
\left\{\begin{array}{l}
x=x(t) \\
y=y(t), \quad t \in\left[t_{1}, t_{2}\right] \\
z=z(t)
\end{array}\right.
$$

## PLAN OF SOLUTION.

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

$$
A=\oint_{\Gamma}(\vec{a}, d \vec{r})=\oint_{\Gamma} P(x, y, z) d x+Q(x, y, z) d y+R(x, y, z) d z
$$

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

$$
\begin{aligned}
& A=\oint_{\Gamma}(\vec{a}, d \vec{r})= \\
& =\int_{t_{1}}^{t_{2}}\left[P(x(t), y(t), z(t)) x^{\prime}(t)+Q(x(t), y(t), z(t)) y^{\prime}(t)+\right. \\
& \left.+R(x(t), y(t), z(t)) z^{\prime}(t)\right] d t
\end{aligned}
$$

We write the answer.