발산 정리

[T] $\text{F}=\langle x,2y,z\rangle ;$ S is the boundary of the tetrahedron in the first octant formed by plane $x+y+z=1.$

For the following exercises, use a CAS along with the divergence theorem to compute the net outward flux for the fields across the given surfaces S.

[T] $\text{F}=\langle y-2x,{x}^{3}-y,{y}^{2}-z\rangle ;$ S is sphere $\{(x,y,z):{x}^{2}+{y}^{2}+{z}^{2}=4\}.$

[T] $\text{F}=\langle x,y,z\rangle ;$ S is the surface of paraboloid $z=4-{x}^{2}-{y}^{2},$ for $z\ge 0,$ plus its base in the xy-plane.

[T] $\text{F}=\langle z-x,x-y,2y-z\rangle ;$ D is the region between spheres of radius 2 and 4 centered at the origin.

[T] $\text{F}=\frac{\text{r}}{||\text{r}||}=\frac{\langle x,y,z\rangle }{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}};$ D is the region between spheres of radius 1 and 2 centered at the origin.

[T] $\text{F}=\langle {x}^{2},-{y}^{2},{z}^{2}\rangle ;$ D is the region in the first octant between planes $z=4-x-y$ and $z=2-x-y.$

Let $\text{F}(x,y,z)=2x\text{i}-3xy\text{j}+x{z}^{2}\text{k}.$ Use the divergence theorem to calculate ${\iint }_{S}\text{F}\cdot d\text{S},$ where S is the surface of the cube with corners at $(0,0,0),(1,0,0),(0,1,0),$ $(1,1,0),(0,0,1),(1,0,1),(0,1,1),\,\text{and}\,(1,1,1),$ oriented outward.

Use the divergence theorem to find the outward flux of field $\text{F}(x,y,z)=({x}^{3}-3y)\text{i}+(2yz+1)\text{j}+xyz\text{k}$ through the cube bounded by planes $x=\pm 1,y=\pm 1,\,\text{and}\,z=\pm 1.$

Let $\text{F}(x,y,z)=2x\text{i}-3y\text{j}+5z\text{k}$ and let S be hemisphere $z=\sqrt{9-{x}^{2}-{y}^{2}}$ together with disk ${x}^{2}+{y}^{2}\le 9$ in the xy-plane. Use the divergence theorem to calculate ${\iint }_{S} F \cdot dS$.

Evaluate ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)={x}^{2}\text{i}+xy\text{j}+{x}^{3}{y}^{3}\text{k}$ and S is the surface consisting of all faces of the tetrahedron bounded by plane $x+y+z=1$ and the coordinate planes, with outward unit normal vector N.

Find the net outward flux of field $\text{F}=\langle bz-cy,cx-az,ay-bx\rangle$ across any smooth closed surface in ${\text{R}}^{3},$ where a, b, and c are constants.

Use the divergence theorem to evaluate ${\iint }_{S}\| \text{R}\| \text{R}\cdot ndS,$ where $\text{R}(x,y,z)=x\text{i}+y\text{j}+z\text{k}$ and S is sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2},$ with constant $a>0.$

Use the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)={y}^{2}z\text{i}+{y}^{3}\text{j}+xz\text{k}$ and S is the boundary of the cube defined by $-1\le x\le 1,-1\le y\le 1,\,\text{and}\,0\le z\le 2.$

Let R be the region defined by ${x}^{2}+{y}^{2}+{z}^{2}\le 1.$ Use the divergence theorem to find ${\iiint }_{R}{z}^{2}dV.$

Let E be the solid bounded by the xy-plane and paraboloid $z=4-{x}^{2}-{y}^{2}$ so that S is the surface of the paraboloid piece together with the disk in the xy-plane that forms its bottom. If $\text{F}(x,y,z)=(xz\,\sin (yz)+{x}^{3})\text{i}+\cos (yz)\text{j}+(3z{y}^{2}-{e}^{{x}^{2}+{y}^{2}})\text{k},$ find ${\iint }_{S}\text{F}\cdot d\text{S}$ using the divergence theorem.

Let E be the solid unit cube with diagonally opposite corners at the origin and (1, 1, 1), and faces parallel to the coordinate planes. Let S be the surface of E, oriented with the outward-pointing normal. Use a CAS to find ${\iint }_{S}\text{F}\cdot d\text{S}$ using the divergence theorem if $\text{F}(x,y,z)=2xy\text{i}+3y{e}^{z}\text{j}+x\,\sin \,z\text{k}.$

Use the divergence theorem to calculate the flux of $\text{F}(x,y,z)={x}^{3}\text{i}+{y}^{3}\text{j}+{z}^{3}\text{k}$ through sphere ${x}^{2}+{y}^{2}+{z}^{2}=1.$

Find ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k}$ and S is the outwardly oriented surface obtained by removing cube $[1,2]\,\times \,[1,2]\,\times \,[1,2]$ from cube $[0,2]\,\times \,[0,2]\,\times \,[0,2].$

Consider radial vector field $\text{F}=\frac{\text{r}}{|\text{r}|}=\frac{\langle x,y,z\rangle }{{({x}^{2}+{y}^{2}+{z}^{2})}^{1/2}}.$ Compute the surface integral, where S is the surface of a sphere of radius a centered at the origin.

Compute the flux of water through parabolic cylinder $S:y={x}^{2},$ from $0\le x\le 2,0\le z\le 3,$ if the velocity vector is $\text{F}(x,y,z)=3{z}^{2}\text{i}+6\text{j}+6xz\text{k}.$