Flux of a vector field

In a steady stationary flow of fluid, the density and velocity at each point of the flow do not depend on time, i.e., $\rho = \rho (x ; y), \mathbf{v} = (u(x ; y) ; v(x ; y))$.

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

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 $x+4 y+z = 4, x \geqslant 0, y \geqslant 0, z \geqslant 0$.

Calculate the integrals.

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 $0 \leqslant x \leqslant a, 0 \leqslant y \leqslant b, 0 \leqslant z \leqslant c$.

Calculate the integrals.

Calculate the integrals.

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

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 $(x-a)^{2}+(y-b)^{2}+(z-c)^{2} = R^{2}$.

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

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

Calculate the integrals.

Calculate the integrals.

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 $\sqrt{y^{2}+z^{2}} \leqslant x \leqslant H$.

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 $x^{2}+y^{2} = z^{2}, 0 < z \leqslant H$.

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

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 $x^{2}+y^{2} = r^{2}, x \leqslant 0, y \geqslant 0,0 \leqslant z \leqslant H$.

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 $z = x^{2}+y^{2}, z \leqslant 1$.

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

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:

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

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

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

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

Find the flux of the field a through the surface $S$, oriented by the normal $\mathbf{n}$, $S(\mathbf{r} = x \mathbf{i}+y \mathbf{j}+z \mathbf{k}, r = \left|\mathbf{r}\right|)$.

Find the flux of the field a through the surface $S$, oriented by the normal $\mathbf{n}$, $S(\mathbf{r} = x \mathbf{i}+y \mathbf{j}+z \mathbf{k}, r = \left|\mathbf{r}\right|)$.

Find the flux of the field a through the surface $S$ either directly or by the Gauss-Ostrogradsky theorem, if:

Let $A(\mathbf{r}) = \sum_{i, j = 1}^{3} a_{i j} x_{i} x_{j}$ be a positive definite quadratic form, $\mathbf{r} = x_{1} \mathbf{i}+x_{2} \mathbf{j}+x_{3} \mathbf{k}$. Find the flux of the field $\mathbf{a} = \mathbf{r} \cdot (A(\mathbf{r}))^{-3 / 2}$ through the unit sphere $\left|\mathbf{r}\right| = 1$.

Give, with accuracy up to $o\left(\varepsilon^{3}\right)$, the approximate value of the flux of the field a:

Let a bounded region $G$ have a piecewise smooth boundary $\partial G$, oriented by the outward normal. Prove that the flux of the radius vector $\mathbf{r}$ through $\partial G$ equals $3 \mu G$, where $\mu G$ is the volume of $G$.

Let the piecewise smooth boundary $\partial G$ of the region $G$ be oriented by the normal $\mathbf{n}, c$ a constant vector. Prove that

Let $S$ be a smooth surface, oriented by the normal $\mathbf{n}$, and suppose the closure of $S$ does not contain the origin. Show that the integral

is the flux of some field through $S$.

Let a bounded region $G$ have a piecewise smooth boundary $\partial G$, oriented by the outward normal, $M_{0}$ a fixed point of $G$,

$S_{\varepsilon }\left(M_{0}\right)$ is the sphere with center $M_{0}$ and radius $\varepsilon$, lying in $G$, oriented by the outward normal. Prove that the flux of a through $\partial G$ equals the flux of $a$ through $S_{\varepsilon }\left(M_{0}\right)$.

Let a bounded region $G$ have a piecewise smooth boundary $\partial G$, oriented by the outward normal, $M_{0}$ a fixed point,

Find the flux of the field a through $\partial G$, if:

Under the conditions of problem 83, let $M_{0} \in \partial G$ and, in a neighborhood of $M_{0}$, the boundary $\partial G$ be twice continuously differentiable. Let $\partial G_{\varepsilon }$ be the part of the boundary $\partial G$ lying inside the ball $\left|\overline{M_{0} M}\right| \leqslant \varepsilon$, and let $\Pi_{\varepsilon }$ be the flux of the field a through $\partial G \backslash \partial G_{\varepsilon }$. Find $\lim_{\varepsilon \rightarrow 0} \Pi_{\varepsilon }$.

Let $\gamma$ be a smooth plane simple (p. 295) curve whose closure does not contain the origin, $\mathbf{n}$ being a continuous unit normal to $\gamma$. Show that the Gauss integral

is the flux of some field through $\gamma$.

Under the conditions of problem $87$, let $\gamma$ be the boundary of a bounded region $G$. Calculate the Gauss integral, if:

Show that the value of the Gauss integral from problem 87 equals the polar angle at which the curve $\gamma$ is seen from the origin.