Surface integrals of the second kind (over oriented surfaces)

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} x\, dy\, dz+y\, dx\, dz+z\, dx\, dy$, where $S$ is the outer surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$.

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} f(x)\, dy\, dz+g(y)\, dx\, dz+h(z)\, dx\, dy$, where $f(x)$, $g(y)$, $h(z)$ are continuous functions and $S$ is the outer surface of the rectangular parallelepiped $0<x<a$, $0<y<b$, $0<z<c$.

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} (y-z)\, dy\, dz+(z-x)\, dx\, dz+(x-y)\, dx\, dy$, where $S$ is the outer surface of the conical surface $x^{2}+y^{2}=z^{2}$ $(0\leqslant z\leqslant h)$.

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} \dfrac {dy\, dz}{x}+\dfrac {dx\, dz}{y}+\dfrac {dx\, dy}{z}$, where $S$ is the outer surface of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$.

Compute the following surface integral of the second kind: $\displaystyle \iint_{S} x^{2}\, dy\, dz+y^{2}\, dx\, dz+z^{2}\, dx\, dy$, where $S$ is the outer surface of the spherical shell $(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=R^{2}$.

Apply the Ostrogradsky formula to transform the following surface integral, where the smooth surface $S$ encloses the bounded volume $V$, and $\cos \alpha$, $\cos \beta$, $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$: $\displaystyle \iint_{S} x^{3}\, dy\, dz+y^{3}\, dx\, dz+z^{3}\, dx\, dy$.

Apply the Ostrogradsky formula to transform the following surface integral, where the smooth surface $S$ encloses the bounded volume $V$, and $\cos \alpha$, $\cos \beta$, $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$: $\displaystyle \iint_{S} xy\, dx\, dy+xz\, dx\, dz+yz\, dy\, dz$.

Apply the Ostrogradsky formula to transform the following surface integral, where a smooth surface $S$ encloses a bounded volume $V$, and $\cos \alpha$, $\cos \beta$, $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$: $\displaystyle \iint_{S} \dfrac {x\cos \alpha +y\cos \beta +z\cos \gamma }{\sqrt{x^{2}+y^{2}+z^{2}}}\, dS$.

Apply the Ostrogradsky formula to transform the following surface integral, where a smooth surface $S$ encloses a bounded volume $V$, and $\cos \alpha$, $\cos \beta$, $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$: $\displaystyle \iint_{S} \left(\dfrac {\partial u}{\partial x}\cos \alpha +\dfrac {\partial u}{\partial y}\cos \beta +\dfrac {\partial u}{\partial z}\cos \gamma \right)\, dS$.

Apply the Ostrogradsky formula to transform the following surface integral, where a smooth surface $S$ encloses a bounded volume $V$, and $\cos \alpha$, $\cos \beta$, $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$:

Using the Ostrogradsky formula, compute the following surface integral: $\displaystyle \iint_{S} x^{2}\, dy\, dz+y^{2}\, dx\, dz+z^{2}\, dx\, dy$, where $S$ is the outer surface of the boundary of the cube $0<x<a$, $0<y<a$, $0<z<a$.

Using the Ostrogradsky formula, compute the following surface integral: $\displaystyle \iint_{S} x^{3}\, dy\, dz+y^{3}\, dx\, dz+z^{3}\, dx\, dy$, where $S$ is the outer surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$.

Using the Ostrogradsky formula, compute the following surface integral: $\displaystyle \iint_{S} (x-y+z)\, dy\, dz+(y-z+x)\, dx\, dz+(z-x+y)\, dx\, dy$, where $S$ is the outer surface of the surface $\left|x-y+z\right|+\left|y-z+x\right|+\left|z-x+y\right|=1$.

Compute

where $S$ is the part of the conical surface $x^{2}+y^{2}=z^{2}$ $(0\leqslant z\leqslant h)$, and $\cos \alpha$, $\cos \beta$, $\cos \gamma$ are the direction cosines of the outward normal of this surface.

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.

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 $S$.

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 $S$ :

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

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 $S$ :

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 $S$ :

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 $S$ :