Flux and circulation of a vector field

Find the flux of the vector $\overrightarrow {r}$: (1) through the lateral surface of the cone $x^{2}+y^{2}\leqslant z^{2}$ $(0\leqslant z\leqslant h)$; (2) through the base of this cone.

Find the flux of the vector $\overrightarrow {a}=yz\overrightarrow {i}+zx\overrightarrow {j}+xy\overrightarrow {k}$: (a) through the lateral surface of the cylinder $x^{2}+y^{2}\leqslant a^{2}$ $(0\leqslant z\leqslant h)$; (б) through the entire surface of this cylinder.

Find the flux of the radius vector $\overrightarrow {r}$ through the surface

Find the flux of the vector $\overrightarrow {a}=x^{2}\overrightarrow {i}+y^{2}\overrightarrow {j}+z^{2}\overrightarrow {k}$ through the part of the sphere $x^{2}+y^{2}+z^{2}=1$, $x\geqslant 0$, $y\geqslant 0$, $z\geqslant 0$, lying in the first octant.

Find the flux of the vector $\overrightarrow {a}=y\overrightarrow {i}+z\overrightarrow {j}+x\overrightarrow {k}$ through the entire surface of the pyramid bounded by the planes $x=0$, $y=0$, $z=0$, $x+y+z=a$ $(a>0)$.

Verify the result using the Ostrogradsky formula.

Prove that the flux of the vector $\overrightarrow {a}$ through the surface $S$ given by the equation $\overrightarrow {r}=\overrightarrow {r}(u,v)$ $((u,v)\in \Omega )$ equals

where $\overrightarrow {n}$ is the unit normal vector of the surface $S$.

Find the flux of the vector $\overrightarrow {a}=m\dfrac {\overrightarrow {r}}{r^{3}}$ ($m$ a constant) through a closed surface $S$ surrounding the origin.

Given the vector

where $e_{i}$ are constants and $r_{i}$ is the distance from the point $M_{i}$ (the starting point) to the moving point $M(\overrightarrow {r})$. Find the flux of this vector through a closed surface $S$ surrounding the points $M_{i}$ $(i=1,2,\ldots ,n)$.

Find the work of the vector $\overrightarrow {a}=\overrightarrow {r}$ along the segment of the helix

Find the work of the vector $\overrightarrow {a}=f(r)\overrightarrow {r}$ (where $f$ is a continuous function) along the arc $AB$.

Find the circulation of the vector

($c$ a constant): (1) along the circle $x^{2}+y^{2}=1$, $z=0$; (2) along the circle $(x-2)^{2}+y^{2}=1$, $z=0$.

Find the circulation $\Gamma$ of the vector $\overrightarrow {a}=\operatorname {grad}\left(\arctan \dfrac {y}{x}\right)$ along a closed contour $C$: (1) $C$ does not encircle the $Oz$ axis; (2) $C$ encircles the $Oz$ axis.