Math Problems Vault

Problem 3.11.1

(1)(2)

Compute the integrals. $\iint_{S}(x+y+z) d S$ , where:

?

(1)

$S$ — the part of the plane $x+2 y+4 z = 4$ , cut out by the conditions $x \geqslant 0, y \geqslant 0, z \geqslant 0$

(2)

$S$ — the part of the sphere $x^{2}+y^{2}+z^{2} = 1$ , cut out by the condition $z \geqslant 0$ .

Problem 3.11.2

(1)(2)

Compute the integrals. $\iint_{S}\left(x^{2}+y^{2}\right) d S$ , where:

?

(1)

$S$ — the sphere $x^{2}+y^{2}+z^{2} = R^{2}$ ;

(2)

$S$ — the surface of the cone $\sqrt{x^{2}+y^{2}} \leqslant z \leqslant 1$ .

Problem 3.11.3

(1)(2)(3)(4)

Compute the integrals. $\iint_{S}\left(x^{2}+y^{2}+z^{2}\right) d S$ , where:

?

(1)

$S$ — the sphere $x^{2}+y^{2}+z^{2} = R^{2}$ ;

(2)

(3)

(4)

$S$ — the total surface of the cylinder $x^{2}+y^{2} \leqslant r^{2}, 0 \leqslant z \leqslant H$ .

Problem 3.11.4

Compute the integrals. $\iint \frac{d S}{(1+x+y)^{2}}, S$ — the surface of the tetrahedron $x+y+z \leqslant 1, x \geqslant 0, y \geqslant 0, z \geqslant 0$ .

?

Problem 3.11.5

(1)(2)

Compute the integrals.

?

(1)

$\iint_{S} x y z d S$ ;

(2)

$\iint_{S}\left|x y\right| z d S$ ; where $S$ — the part of the paraboloid $z = x^{2}+y^{2}$ , cut out by the condition $z \leqslant 1$ .

Problem 3.11.6

(1)(2)

Compute the integrals.

?

(1)

$\iint_{S}\left(x^{2}+y^{2}\right) d S$ ;

(2)

$\iint_{S} \sqrt{x^{2}+y^{2}} d S$ ; where $S$ — the part of the conical surface $z = \sqrt{x^{2}+y^{2}}$ , cut out by the condition $z \leqslant 1$ .

Problem 3.11.7

(1)(2)

Compute the integrals.

?

(1)

$\iint_{S}(x y+y z+z x) d S$ ;

(2)

$\iint_{S}\left(x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2}\right) d S$ ; where $S$ — the part of the conical surface $z = \sqrt{x^{2}+y^{2}}$ , lying inside the cylinder $x^{2}+y^{2} = 2 x$ .

Problem 3.11.8

(1)(2)(3)

Calculate the integrals.

?

(1)

$\iint_{S} f(x ; y ; z) d S$ ;

(2)

$\iint_{S} \frac{d S}{f(x ; y ; z)}$ ;

(3)

$\iint_{S}\left(x^{2}+y^{2}+z^{2}\right)^{-3 / 2} \frac{d S}{f(x ; y ; z)}$ ; where $f = \sqrt{\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}+\frac{z^{2}}{c^{4}}}, S$ is the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1$ .

Problem 3.11.9

Calculate the integrals. $\iint_{S}\left(x^{2}+y^{2}+(z-a)^{2}\right)^{-n / 2} d S, \quad n \in N, \quad S$ — the sphere $x^{2}+y^{2} + z^{2} = R^{2}$ .

?

Problem 3.11.10

Calculate the integrals. $\iint_{S} z^{2} d S, S$ — part of the conical surface $x = u \cos v \sin \alpha , \quad y = u \sin v \sin \alpha , \quad z = u \cos \alpha$ ,

$\alpha = \mathrm{const}, \alpha \in (0 ; \pi / 2)$ , singled out by the conditions $u \in [0 ; 1], v \in [0 ; 2 \pi ]$ .

?

Problem 3.11.11

Calculate the integrals. $\iint_{S} z d S, S$ is the surface

x = u \cos v, \quad y = u \sin v, \quad z = v, \quad u \in [0 ; 1], \quad v \in [0 ; 2 \pi ]

?

Problem 3.11.12

Calculate the integrals. $\iint_{S} f(r) d S$ , where $r = \sqrt{x^{2}+y^{2}+z^{2}}, \quad f(r) = \begin{cases} 1-r^{2}, & r \leqslant 1, \\ 0, & r \geqslant 1, \end{cases} S$ is the plane $x+y+z = a$ .

?

Problem 3.11.13

Calculate the integrals. $\iint_{S} f(r ; z) d S$ , where $r = \sqrt{x^{2}+y^{2}}, \quad f(r ; z) = \begin{cases} r^{2}, & r \leqslant z, \\ 0, & r \geqslant z, \end{cases} \quad S-$ the sphere $x^{2}+y^{2}+z^{2} = R^{2}$ .

?

Problem 3.11.14

Prove the Poisson formula

\iint _{S} f(a x+b y+c z) d S = 2 \pi \int _{-1}^{1} f\left(\sqrt{a^{2}+b^{2}+c^{2}} t\right) d t

where $f(t),\left|t\right| \leqslant \sqrt{a^{2}+b^{2}+c^{2}}$ , is a continuous function, $S$ is the sphere $x^{2}+y^{2}+z^{2} = 1$ .

?

Problem 3.11.15

(1)(2)(3)(4)

Determine the mass distributed:

?

(1)

over the surface of the cube $0 \leqslant x \leqslant a, 0 \leqslant y \leqslant a, 0 \leqslant z \leqslant a$ with surface density $\rho = \rho_{0} x y z$ ;

(2)

over the sphere $x^{2}+y^{2}+z^{2} = R^{2}$ with density: а) $\rho = \rho_{0} \sqrt{x^{2}+y^{2}}$ , б) $\rho = \rho_{0}\left(x^{2}+y^{2}\right)$ ;

(3)

over part of the elliptic paraboloid $x^{2}+y^{2} = 2 z, z \leqslant 1$ with density $\rho = \rho_{0} z$ ;

(4)

over part of the hyperbolic paraboloid $x^{2}-y^{2} = 2 z$ , cut out by the cylinder $x^{2}+y^{2} = 1$ , with density $\rho = \rho_{0}\left|z\right|, \rho_{0} =$ const.

Problem 3.11.16

(1)(2)

Determine the static moment relative to the plane $z = 0$ of a homogeneous ( $\rho = \rho_{0} =$ const $)$ surface:

?

(1)

$x+y+z = a, x \geqslant 0, y \geqslant 0, z \geqslant 0$ ;

(2)

$x^{2}+y^{2}+z^{2} = R^{2}, z \geqslant 0$ .

Problem 3.11.17

(1)(2)(3)

Determine the applicate (z-coordinate) of the center of mass of the hemisphere $x^{2}+y^{2}+z^{2} = R^{2}, z \geqslant 0$ with surface density:

?

(1)

$\rho = \rho_{0}$ ;

(2)

$\rho = \rho_{0} \sqrt{x^{2}+y^{2}}$ ;

(3)

$\rho = \rho_{0}\left(x^{2}+y^{2}\right), \rho_{0} =$ const.

Problem 3.11.18

Determine the coordinates of the center of mass of homogeneous surfaces:

?

(1)

$x^{2}+y^{2}+z^{2} = R^{2}, x \geqslant 0, y \geqslant 0, z \geqslant 0$

(2)

$z = \sqrt{R^{2}-x^{2}-y^{2}}, x \geqslant 0, y \geqslant 0, x+y \leqslant R$ ;

(3)

$z = \sqrt{x^{2}+y^{2}}, x^{2}+y^{2} \leqslant x$ ;

(4)

$z = 2-\left(x^{2}+y^{2}\right) / 2, z \geqslant 0$ ;

(5)

$x = u \cos v, y = u \sin v, z = v, u \in [0 ; 1], v \in [0 ; \pi ]$ .

Problem 3.11.19

(1)(2)

Calculate the moments of inertia relative to the coordinate planes of a homogeneous ( $\rho = \rho_{0} =$ const $)$ surface:

?

(1)

$x+y+z = 1, x \geqslant 0, y \geqslant 0, z \geqslant 0$

(2)

$z = \frac{h}{r} \sqrt{x^{2}+y^{2}}, x^{2}+y^{2} \leqslant r^{2}$ .

Problem 3.11.20

(1)(2)

Calculate the moment of inertia of a homogeneous ( $\rho = \rho_{0} =$ const) surface:

?

(1)

$x^{2}+y^{2} = 2 a z, z \leqslant a$ , relative to the axis $O z$ ;

(2)

$x^{2} / a^{2}+y^{2} / a^{2} = z^{2} / b^{2}, 0 \leqslant z \leqslant b$ , relative to the line $y = 0$ , $z = b$ .

Problem 3.11.21

(1)(2)

Find the magnitude of the force with which a homogeneous surface:

?

(1)

$x = a \cos \varphi , y = a \sin \varphi , z = z, \varphi \in [0 ; 2 \pi ], z \in [0 ; H]$ ;

(2)

$x = r \cos \varphi , y = r \sin \varphi , z = r, \varphi \in [0 ; 2 \pi ], r \in [a ; b], a > 0$ ; of density $\rho_{0}$ , attracts a point mass $m$ placed at the origin.

Problem 3.11.22

Find the magnitude of the force with which a homogeneous sphere of radius $R$ and density $\rho = \rho_{0}$ attracts a point mass $m$ .

?

Problem 3.11.23

(1)(2)

Determine the electric charge distributed with density $\rho = \rho_{0}\left|z\right|$ over the surface:

?

(1)

$x^{2} / a^{2}+y^{2} / a^{2}-z^{2} / c^{2} = 0,\left|z\right| \leqslant c ;$

(2)

$z^{2}-x^{2}-y^{2} = a^{2},\left|z\right| \leqslant a \sqrt{2}$ .

Problem 3.11.24

(1)(2)

Find the potential at the point $M_{0}\left(x_{0} ; y_{0} ; z_{0}\right)$ of a simple layer (item 1), distributed:

?

(1)

on the sphere $x^{2}+y^{2}+z^{2} = R^{2}$ with constant density $\mu_{0}$ ;

(2)

on the sphere $x^{2}+y^{2}+z^{2} = R_{1}^{2}$ with constant density $\mu_{1}$ and on the sphere $x^{2}+y^{2}+z^{2} = R_{1}^{2}$ with constant density $\mu_{2}, R_{1} < R_{2}$ .

Problem 3.11.25

(1)(2)

Find, at the point $(0 ; 0 ; z)$ , the potential of a simple layer distributed with density $\mu$ :

?

(1)

on the lateral surface of the cylinder $x^{2}+y^{2} = R^{2}, 0 \leqslant z \leqslant H$ , $\mu = \mu_{0}$ ;

(2)

on the sphere $x^{2}+y^{2}+z^{2} = R^{2}, \mu = \mu_{0} z^{2}$ .

Problem 3.11.26

Calculate the integrals. $\iint_{S}\left(x^{2}+y^{2}\right) d x d y, S$ — the lower side of the disk $x^{2}+y^{2} \leqslant 4$ , $z = 0$ .

?

Problem 3.11.27

Calculate the integrals. $\iint_{S}(2 z-x) d y d z+(x+2 z) d z d x+3 z d x d y, S$ — the upper side of the triangle $x+4 y+z = 4, x \geqslant 0, y \geqslant 0, z \geqslant 0$ .

?

Problem 3.11.28

(1)(2)

Calculate the integrals.

?

(1)

$\iint_{S} x z d x d y$ ;

(2)

$\iint_{S} y z d y d z+z x d z d x+x y d x d y ; \quad S$ the inner side of the surface of the tetrahedron $x+y+z \leqslant 1, x \geqslant 0$ , $y \geqslant 0, z \geqslant 0$ .

Problem 3.11.29

Calculate the integrals. $\iint_{S} f_{1}(x) d y d z+f_{2}(y) d z d x+f_{3}(z) d x d y$ , where $f_{1}, f_{2}, f_{3}$ are continuous functions, $S$ is the outer side of the surface of the parallelepiped $0 \leqslant x \leqslant a, 0 \leqslant y \leqslant b, 0 \leqslant z \leqslant c$ .

?

Problem 3.11.30

(1)(2)

Calculate the integrals.

?

(1)

$\iint_{S} y d z d x$ ;

(2)

$\iint_{S} x^{2} d y d z ; S$ — the outer side of the sphere $x^{2}+y^{2}+z^{2} = R^{2}$ .

Problem 3.11.31

(1)(2)

Calculate the integrals.

?

(1)

$\iint_{S}\left(x^{5}+z\right) d y d z$ ;

(2)

$\iint_{S} x^{2} y^{2} z d x d y ; S$ — the inner side of the hemisphere $x^{2}+y^{2}+z^{2} = R^{2}, z \leqslant 0$ .

Problem 3.11.32

Calculate the integrals. $\iint_{S} x^{2} d y d z+z^{2} d x d y, \quad S$ — the outer side of part of the sphere $x^{2}+y^{2}+z^{2} = R^{2}, x \leqslant 0, y \geqslant 0$ .

?

Problem 3.11.33

Calculate the integrals. $\iint_{S} x^{2} d y d z+y^{2} d z d x+z^{2} d x d y, S$ — the outer side of the sphere $(x-a)^{2}+(y-b)^{2}+(z-c)^{2} = R^{2}$ .

?

Problem 3.11.34

Calculate the integrals. $\iint z^{2} d x d y, S$ — the inner side of the hemisphere $(x-a)^{2} + (y-b)^{2}+z^{2} = R^{2}, z \geqslant 0$ .

?

Problem 3.11.35

Calculate the integrals. $\iint_{S}(x-1)^{3} d y d z, S$ — the outer side of the hemisphere $x^{2}+y^{2} + z^{2} = 2 x, z \leqslant 0$ .

?

Problem 3.11.36

(1)(2)(3)(4)

Calculate the integrals.

?

(1)

$\iint_{S} d z d x$ ;

(2)

$\iint_{S} x d y d z$ ;

(3)

$\iint_{S} x^{2} d y d z$ ;

(4)

$\iint_{S} \frac{d x d y}{z} ; S$ — the outer side of the ellipsoid $x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2} = 1$ .

Problem 3.11.37

(1)(2)

Calculate the integrals.

?

(1)

$\iint_{S} y z d z d x$ ;

(2)

$\iint_{S} x^{3} d y d z+y^{3} d z d x ; S$ — the outer side of part of the ellipsoid $x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2} = 1, z \geqslant 0$ .

Problem 3.11.38

Calculate the integrals. $\iint_{S}\left(2 x^{2}+y^{2}+z^{2}\right) d y d z, S$ — the outer side of the lateral surface of the cone $\sqrt{y^{2}+z^{2}} \leqslant x \leqslant H$ .

?

Problem 3.11.39

Calculate the integrals. $\iint_{S}(y-z) d y d z+(z-x) d z d x+(x-y) d x d y, S$ — one of the sides of the surface $x^{2}+y^{2} = z^{2}, 0 < z \leqslant H$ .

?

Problem 3.11.40

Calculate the integrals. $\iint_{S} y z^{2} d x d z, S$ — the inner side of part of the cylindrical surface $x^{2}+y^{2} = r^{2}, y \leqslant 0,0 \leqslant z \leqslant r$ .

?

Problem 3.11.41

Calculate the integrals. $\iint_{S} y z d x d y+z x d y d z+x y d z d x, S$ — the outer side of part of the cylinder $x^{2}+y^{2} = r^{2}, x \leqslant 0, y \geqslant 0,0 \leqslant z \leqslant H$ .

?

Problem 3.11.42

Calculate the integrals. $\iint_{S} x^{6} d y d z+y^{4} d z d x+z^{2} d x d y, S$ — the lower side of part of the elliptic paraboloid $z = x^{2}+y^{2}, z \leqslant 1$ .

?

Problem 3.11.43

Calculate the integrals. $\iint_{S} x d y d z+y d z d x+z d x d y, S$ — the upper side of part of the hyperbolic paraboloid $z = x^{2}-y^{2},\left|y\right| \leqslant x \leqslant a$ .

?

Problem 3.11.44

(1)(2)

Using the Gauss-Ostrogradsky theorem, calculate the integrals. $\iint_{S}(1+2 x) d y d z+(2 x+3 y) d z d x+(3 y+4 z) d x d y$ , where $S$ :

?

(1)

the outer side of the surface of the pyramid $x / a+y / b+z / c \leqslant 1$ , $x \geqslant 0, y \geqslant 0, z \geqslant 0$

(2)

Problem 3.11.45

(1)(2)(3)

Using the Gauss-Ostrogradsky theorem, calculate the integrals. $\iint_{S} z d x d y+(5 x+y) d y d z$ , where $S$ :

?

(1)

the outer side of the total surface of the cone $x^{2}+y^{2} \leqslant z^{2}$ , $0 \leqslant z \leqslant 4$

(2)

the inner side of the ellipsoid $x^{2} / 4+y^{2} / 9+z^{2} = 1$ ;

(3)

the outer side of the boundary of the region $1 < x^{2}+y^{2}+z^{2} < 4$ .

Problem 3.11.46

(1)(2)

Using the Gauss-Ostrogradsky theorem, calculate the integrals. $\iint_{S} x^{2} d y d z+y^{2} d z d x+z^{2} d x d y$ , where $S$ :

?

(1)

the inner side of the surface of the parallelepiped $0 \leqslant x \leqslant a$ , $0 \leqslant y \leqslant b, 0 \leqslant z \leqslant c ;$

(2)

the outer side of the total surface $x^{2} / a^{2}+y^{2} / b^{2} \leqslant z^{2} / c^{2}, 0 \leqslant z \leqslant c$ (cone).

Problem 3.11.47

(1)(2)

Using the Gauss-Ostrogradsky theorem, calculate the integrals. $\iint_{S} x^{3} d y d z+y^{3} d z d x+z^{3} d x d y$ , where $S$ :

?

(1)

the outer side of the surface of the tetrahedron $x+y+z \leqslant a, x \geqslant 0$ , $y \geqslant 0, z \geqslant 0$

(2)

the inner side of the sphere $x^{2}+y^{2}+z^{2} = R^{2}$ .

Problem 3.11.48

(1)(2)

Using the Gauss-Ostrogradsky theorem, calculate the integrals. $\iint_{S} x^{4} d y d z+y^{4} d z d x+z^{4} d x d y$ , where $S$ :

?

(1)

the sphere $x^{2}+y^{2}+z^{2} = R^{2}$ ;

(2)

the outer side of the total surface of the hemisphere $x^{2}+y^{2}+z^{2} \leqslant R^{2}, z \geqslant 0$ .

Problem 3.11.49

Prove the formula for the volume $V$ of a body bounded by a smooth surface $S$

V = \left|\frac{1}{3} \iint _{S} x d y d z+y d z d x+z d x d y\right|

?

Problem 3.11.50

(1)(2)(3)

Using the formula from problem 49, find the volume of the body bounded by:

?

(1)

the surface $x = u \cos v, y = u \sin v, z = -u+a \cos v(u \geqslant 0$ , $a > 0)$ and the planes $x = 0, z = 0$ ;

(2)

the surface $x = (b+a \cos u) \cos v, y = (b+a \cos u) \sin v ; z = a \sin u, b \geqslant a > 0$

(3)

the surface $x = a \cos u \cos v+b \sin u \sin v, y = a \cos u \sin v- -b \sin u \cos v, z = c \sin u$ and the planes $z = c, z = -c$ .

Problem 3.11.51

(1)(2)

Calculate the integrals. $\iint_{S} x d y d z+y d z d x+z d x d y$ , where $S$ is the outer side of the surface formed by rotating about the axis $z$ the curve:

?

(1)

$y = 2-\left|z-1\right|, z \in [0 ; 2]$ ;

(2)

$x = 1+\sin z, z \in [0 ; \pi ]$ .

Problem 3.11.52

(1)(2)(3)

Calculate the integrals. $\iint_{S} x^{2} d y d z+y^{2} d z d x+z^{2} d x d y$ , where $S$ :

?

(1)

the lower side of the hemisphere $x^{2}+y^{2}+z^{2} = R^{2}, z \geqslant 0$ ;

(2)

the upper side of part of the surface of the paraboloid $x^{2}+y^{2} + 2 a z = a^{2}, z \geqslant 0$

(3)

the lower side of part of the conical surface $x^{2}+y^{2} = z^{2}$ , $0 < z \leqslant H$ .

Problem 3.11.53

Calculate the integrals. $\iint_{S} \left(z^{2}-y^{2}\right) d y d z+\left(x^{2}-z^{2}\right) d z d x+\left(y^{2}-x^{2}\right) d x d y$ , $S$ — the upper side of the hemisphere $x^{2}+y^{2}+z^{2} = R^{2}, z \geqslant 0$ .

?

Problem 3.11.54

Calculate the integrals. $\iint_{S} x^{2} y d y d z+x y^{2} d z d x+x y z d x d y, \quad S$ — the lower side of part of the sphere $x^{2}+y^{2}+z^{2} = R^{2}, x \geqslant 0, y \geqslant 0, z \geqslant 0$ .

?

Problem 3.11.55

Calculate the integrals. $\iint_{S} x^{2} y d y d z-x y^{2} d z d x+\left(x^{2}+y^{2}\right) z d x d y, S$ — the outer side of part of the cylindrical surface $x^{2}+y^{2} = R^{2}, 0 \leqslant z \leqslant H$ .

?

Problem 3.11.56

Prove that if $S$ is a closed smooth surface, $\mathbf{n}$ its outward normal, $\mathbf{l}$ some constant vector, then

\iint _{S} \cos (\widehat{\mathbf{l}, \mathbf{n}}) d S = 0

?

Problem 3.11.57

(1)(2)

Let $G \in R^{3}$ be a bounded domain with smooth boundary $S$ , $\mathbf{n}$ the outward normal to $S, \mathbf{r} = (\xi -x) \mathbf{i}+(\eta -y) \mathbf{j}+(\zeta -z) \mathbf{k}$ :

?

(1)

prove the formula

\iint _{S} \cos (\widehat{\mathbf{r}, \mathbf{n}}) d S = 2 \iiint _{G} \frac{d \xi \, d \eta \, d \zeta }{\left|\mathbf{r}\right|}

(2)

calculate the Gauss integral

I(x ; y ; z) = \iint _{S} \frac{\cos (\widehat{\mathbf{r}, \mathbf{n}})}{\mathbf{r}^{2}} d S, \quad (x ; y ; z) \notin S

Problem 3.11.58

Prove that if $G \in R^{3}$ is a bounded domain with smooth boundary $S, \mathbf{n}$ the outward normal to $S, u(x ; y ; z)$ and $v(x ; y ; z)$ are functions twice continuously differentiable on $\bar{G}$ , then

\iiint _{G}\left[\begin{smallmatrix} \Delta u & \Delta v \\ u & v \end{smallmatrix}\right] d x d y d z = \iint _{S}\left[\begin{smallmatrix} \frac{\partial u}{\partial \mathbf{n}} & \frac{\partial v}{\partial \mathbf{n}} \\ u & v \end{smallmatrix}\right] d S.

?

Problem 3.11.59

Prove that if $u(x ; y ; z)$ is a harmonic function in the bounded closed domain $\bar{G}$ with smooth boundary $S, \mathbf{n}$ the outward normal to $S, \mathbf{r} = (\xi -x) \mathbf{i}+(\eta -y) \mathbf{j}+(\zeta -z) \mathbf{k}$ , then

u(x ; y ; z) = \frac{1}{4 \pi } \iint _{S}\left(u \frac{\cos (\widehat{\mathbf{r}, \mathbf{n}})}{\mathbf{r}^{2}}+\frac{1}{\left|\mathbf{r}\right|} \frac{\partial u}{\partial \mathbf{n}}\right) d S

?

Problem 3.11.60

Prove that if $u(x ; y ; z)$ is a function harmonic inside the sphere $S$ of radius $R$ centered at the point $\left(x_{0} ; y_{0} ; z_{0}\right)$ , then

u\left(x_{0} ; y_{0} ; z_{0}\right) = \frac{1}{4 \pi R^{2}} \iint _{S} u(x ; y ; z) d S

?

Problem 3.11.61

Using Stokes' formula, calculate the integrals. $\int_{L}(x+z) d x+(x-y) d y+x d z, L$ — the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1, z = c$ , oriented negatively with respect to the vector $(0 ; 0 ; 1)$ .

?

Problem 3.11.62

Using Stokes' formula, calculate the integrals. $\int_{L} y^{2} d x+z^{2} d y+x^{2} d z, L$ — the boundary of the triangle with vertices at the points $(a ; 0 ; 0),(0 ; a ; 0),(0 ; 0 ; a)$ , oriented positively with respect to the vector $(0 ; 1 ; 0)$ .

?

Problem 3.11.63

(1)(2)

Using Stokes' formula, calculate the integrals.

?

(1)

$\int_{L} y d x+z d y+x d z$ ;

(2)

$\int_{L} \frac{x d y-y d x}{x^{2}+y^{2}}+z d z$ ; where $L$ is the circle $x^{2}+y^{2}+z^{2} = R^{2}$ , $x+y+z = 0$ , oriented positively with respect to the vector $(0 ; 0 ; 1)$ .

Problem 3.11.64

Using Stokes' formula, calculate the integrals. $\int_{L}\left(y^{2}-z^{2}\right) d x+\left(z^{2}-x^{2}\right) d y+\left(x^{2}-y^{2}\right) d z, L$ — the curve of intersection of the surface of the cube $\left|x\right| \leqslant a,\left|y\right| \leqslant a,\left|z\right| \leqslant a$ with the plane $x + y+z = 3 a / 2$ , oriented positively with respect to the vector $(1 ; 0 ; 0)$ .

?

Problem 3.11.65

(1)(2)

Using Stokes' formula, calculate the integrals. $\int_{L}(y-z) d x+(z-x) d y+(x-y) d z$ , where:

?

(1)

$L$ is the circle $x^{2}+y^{2}+z^{2} = R^{2}, y = x \operatorname {tg} \varphi , \varphi \in (0 ; \pi )$ , oriented positively with respect to the vector $(1 ; 0 ; 0)$ ;

(2)

$L$ is the ellipse $x^{2}+y^{2} = a^{2}, x / a+z / c = 1, a > 0, c > 0$ , oriented negatively with respect to the vector $(1 ; 0 ; 0)$ .

Problem 3.11.66

Using Stokes' formula, calculate the integrals. $\int_{L} y d x-z d y+x d z, L$ — the curve $x^{2}+y^{2}+2 z^{2} = 2 a^{2}, y-x = 0$ , oriented positively with respect to the vector $(1 ; 0 ; 0)$ .

?

Problem 3.11.67

Using Stokes' formula, calculate the integrals. $\int_{L}\left(y^{2}+z^{2}\right) d x+\left(z^{2}+x^{2}\right) d y+\left(x^{2}+y^{2}\right) d z, L$ — the curve $x^{2} + y^{2}+z^{2} = 2 a x, x^{2}+y^{2} = 2 b x, z > 0,0 < b < a$ , oriented positively with respect to the vector $(0 ; 0 ; 1)$ .

?

Problem 3.11.68

Using Stokes' formula, calculate the integrals. $\int_{L} z^{3} d x+x^{3} d y+y^{3} d z, L$ — the curve $2 x^{2}-y^{2}+z^{2} = a^{2}, x+y = 0$ , oriented positively with respect to the vector $(1 ; 0 ; 0)$ .

?

Problem 3.11.69

Calculate the integrals, if the curve $L$ is oriented in the direction of increasing parameter $t$ . $\int_{L} x d x+(x+y) d y+(x+y+z) d z, L$ — the curve $x = a \sin t$ , $y = a \cos t, z = a(\sin t+\cos t), t \in [0 ; 2 \pi )$ .

?

Problem 3.11.70

Calculate the integrals, if the curve $L$ is oriented in the direction of increasing parameter $t$ . $\int_{L} y^{2} z^{2} d x+x^{2} z^{2} d y+x^{2} y^{2} d z, L$ — the curve $x = a \cos t, y = a \cos 2 t, z = a \cos 3 t, t \in [0 ; 2 \pi )$ .

?

Problem 3.11.71

Calculate the integrals, if the curve $L$ is oriented in the direction of increasing parameter $t$ . $\int_{L}(y+z) d x+(z+x) d y+(x+y) d z, L$ — the curve $x = a \sin^{2} t$ , $y = a \sin 2 t, z = a \cos^{2} t, t \in [0 ; \pi )$ .

?

Problem 3.11.72

Calculate the integrals, if the curve $L$ is oriented in the direction of increasing parameter $t$ . $\int_{L}\left(x^{2}-y z\right) d x+\left(y^{2}-z x\right) d y+\left(z^{2}-x y\right) d z, L$ — the curve $x = a \cos t, y = a \sin t, z = h t /(2 \pi ), t \in [0 ; 2 \pi ]$ .

?