Example. Find the modulus of the circulation of the vector field

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

along the closed contour

$$
\Gamma=\left\{(x, y, z): \begin{array}{l}
x^{2}+y^{2}+z^{2}=9 \\
x^{2}+y^{2}=9
\end{array}\right\}
$$
SOLUTION.

1. In this case, it is obvious that $\Gamma$ is the circle $x^{2}+y^{2}=9$ lying in the plane $z=0$. We choose the direction of traversal of the contour $\Gamma$ counterclockwise when viewed from the end of the vector $\vec{k}$.
2. We choose the surface $\Sigma$ stretched over the contour $\Gamma$.

Naturally, as $\Sigma$ we take the circle:

$$
\Sigma=\left\{(x, y, z): \begin{array}{l}
z=0 \\
x^{2}+y^{2} \leq 9
\end{array}\right\}
$$

3. According to the chosen orientation of the contour, the normal $\overrightarrow{n_{0}}$ to $\Sigma$ is the same at every point and equals $\vec{k}$.
4. We find $\operatorname{rot} \vec{a}:$

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

5. We compute the scalar product:

$$
\left(\operatorname{rot} \vec{a}, \overrightarrow{n_{0}}\right)=(2 x \vec{i}-y \vec{j}+(-1-z) \vec{k}, \vec{k})=-1-z
$$

6. We find the circulation using Stokes' formula (1):

$$
A=\iint_{\Sigma}(-1-z) d \sigma
$$

7. We compute the surface integral, reducing it to a double integral:

$$
A=\iint_{\Sigma}(-1-z) d \sigma=\left.\iint_{x^{2}+y^{2} \leq 9}(-1-z)\right|_{z=0} d x d y=-9 \pi
$$

Answer: $|A|=9 \pi$.

PROBLEM CONDITIONS. Find the modulus of the circulation of the vector field $\vec{a}$ along the closed contour $\Gamma$.

1. $\quad \vec{a}=(x z+y) \vec{i}+(y z-x) \vec{j}, \quad \Gamma=\left\{\begin{array}{l}x^{2}+y^{2}=1, \\ z=3\end{array}\right\}$.
2. $\quad \vec{a}=x^{2} z \vec{i}-y \vec{j}+y \vec{k}, \quad \Gamma=\left\{\begin{array}{l}z=3\left(x^{2}+y^{2}\right)-1, \\ z=2\end{array}\right\}$.
3. $\quad \vec{a}=x z \vec{i}+2 x z \vec{j}+x^{3} y \vec{k}, \quad \Gamma=\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}=9, \\ x^{2}+y^{2}=1(z>0)\end{array}\right\}$.
4. $\quad \vec{a}=(x-y) \vec{i}+x \vec{j}-z \vec{k}, \quad \Gamma=\left\{\begin{array}{l}x^{2}+y^{2}=1, \\ z=2\end{array}\right\}$.
5. $\vec{a}=y \vec{i}-x \vec{j}+x z \vec{k}$,

$\Gamma=\left\{\begin{array}{l}z=2\left(x^{2}+y^{2}\right)+1 \\ z=3\end{array}\right\}$
6. $\vec{a}=4 y \vec{i}+x \vec{j}+y \vec{k}$,

$\Gamma=\left\{\begin{array}{l}x^{2}+y^{2}=1 \\ x+y+z=1\end{array}\right\}$
7. $\vec{a}=y \vec{i}-x \vec{j}+z \vec{k}$,

$\Gamma=\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}=1 \\ x=z\end{array}\right\}$
8. $\vec{a}=x y \vec{i}+y z \vec{j}+x z \vec{k}$,

$\Gamma=\left\{\begin{array}{l}x^{2}+y^{2}=4, \\ 2 x+y+z=1\end{array}\right\}$.
9. $\quad \vec{a}=y \vec{i}+(1-x) \vec{j}-z \vec{k}$, $\Gamma=\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}=9, \\ x^{2}+y^{2}=4(z>0)\end{array}\right\}$.
10. $\vec{a}=x \vec{i}-z^{2} \vec{j}+y \vec{k}$, $\Gamma=\left\{\begin{array}{l}x^{2}+y^{2}=4, \\ 4 x-3 y-z=3\end{array}\right\}$.

Answers. 1. $2 \pi$. 2. 0. 3. $4 \sqrt{2} \pi$. 4. $2 \pi$. 5. $2 \pi$. 6. $2 \pi$. 7. $\sqrt{2} \pi$. 8. $8 \pi$. 9. $8 \pi$. 10. $60 \pi$.

Educational Publication

O.V. ZIMINA, A.I. KIRILLOV, T.A. SALNIKOVA

# SOLUTION MANUAL HIGHER MATHEMATICS

Editor E.Yu. Khodan. Computer typesetting by T.A. Salnikova. Computer layout by O.B. Zimina.

LR No. 071930 dated 06.07.99. Signed for printing 02.04.01.

Format 60 x 90 / 16. Paper offset. Printing offset.

Printed sheets 23. Text sheets 22. Print run 5000 copies. Order No. 1490

Publishing firm "Physics and Mathematics Literature"

MAIK "Nauka/Interperiodica"

117864 Moscow, Profsoyuznaya St., 90

Printed from ready-made diapositives

at PTP "Printing House 'Nauka'"

121099 Moscow, Shubinsky Lane, 6

## O.V. ZIMINA, A.I. KIRILLOV, T.A. SALNIKOVA HIGHER MATHEMATICS

## PHYSICS AND MATHEMATICS PUBLISHERS

International Academic Publishing Company "Nauka" Russian Academy of Sciences

Moscow, 2001, 368 pages

The authors have extensive experience in teaching mathematics at the Moscow Institute of Power Engineering (Technical University) and other technical universities in Russia and abroad.

A well-known expert in higher education and a member of the Presidium of the Scientific Methods Council on Mathematics of the Russian Ministry of Education, Professor A.I. Kirillov is a coauthor of every single mathematics curriculum developed and used in Russia during the last twenty years. He is the recipient of several state awards for his successes in teaching higher mathematics to students in technical universities and in using computers in education. He has presented his ideas on mathematical education at many international congresses and conferences and in lectures at many universities around the world and is also known in scientific circles for his significant papers on infinite-dimensional analysis, stochastic processes, and quantum field theory.

THE SOLVER is a part of a new educational project EduXXI initiated by A. I. Kirillov and O. V. Zimina in 1999. The project aim is to make the student's studies effective, interesting, and arousing creative activity. The main means for achieving the project aim are new tutorial complexes consisting of especially written textbooks and software. THE SOLVER is the common name of the family of such complexes.

THE SOLVER "Higher Mathematics" is the first complex realized by the project EduXXI. It is designed for computer-based mathematical education and consists of this book and the software package THE SOLVER.HM.

The complex gives a solution to the fundamental problem of modern education: how can students manage with the enormous information the professors have to give them?

They cannot now. Therefore, the modern students learn almost the same mathematics as the students at the end of the 19th century. Moreover, they solve only one or two problems of each kind whereas they should solve from three to five problems of each kind to really understand and acquire the solution methods.

There are no books containing the necessary number of problems of one kind. On the other