6.8 The divergence theorem  (Page 8/12)

Use the divergence theorem to compute flux integral ${\displaystyle {\iint }_S\text{F}·d\text{S}},$ where $\text{F}(x,y,z)=x+y\text{j}+z^4\text{k}$ and S is a part of cone $z=\sqrt{x^2+y^2}$ beneath top plane $z=1,$ oriented downward.

Use the divergence theorem to calculate surface integral ${\displaystyle {\iint }_S\text{F}·d\text{S}}$ for $\text{F}(x,y,z)=x^4\text{i}-x^3{z}^2\text{j}+4x{y}^{2}z\text{k},$ where S is the surface bounded by cylinder $x^2+y^2=1$ and planes $z=x+2\text{and}z=0.$

Consider $\text{F}(x,y,z)=x^2\text{i}+xy\text{j}+(z+1)\text{k}.$ Let E be the solid enclosed by paraboloid $z=4-x^2-y^2$ and plane $z=0$ with normal vectors pointing outside E . Compute flux F across the boundary of E using the divergence theorem.

[T] $\text{F}=⟨x,\mathrm{-2}y,3z⟩;$ S is sphere $\left\{\left(x,y,z\right):x^2+y^2+z^2=6\right\}.$

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

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

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

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

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

[T] $\text{F}=⟨x^2,\text{−}{y}^2,z^2⟩;$ 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 ${\displaystyle {\iint }_S\text{F}·d\text{S}},$ 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 $\text{F}(x,y,z)=\left(x^3-3y\right)\text{i}+\left(2yz+1\right)\text{j}+xyz\text{k}$ through the cube bounded by planes $x=\mathrm{\pm 1},y=\mathrm{\pm 1},\text{and}z=\mathrm{\pm 1}.$

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.

Evaluate ${\displaystyle {\iint }_S\text{F}·\text{N}dS},$ 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 except 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 $\text{F}=⟨bz-cy,cx-az,ay-bx⟩$ across any smooth closed surface in ${\text{R}}^3,$ where a , b , and c are constants.

Use the divergence theorem to evaluate ${\displaystyle {\iint }_S\Vert \text{R}\Vert \text{R}·n}ds,$ 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 ${\displaystyle {\iint }_S\text{F}·d\text{S}},$ 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 $\mathrm{-1}\le x\le 1,\mathrm{-1}\le y\le 1,\text{and}0\le z\le 2.$

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

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 $\text{F}(x,y,z)=\left(xz\text{sin}(yz)+x^3\right)\text{i}+\text{cos}\left(yz\right)\text{j}+\left(3z{y}^2-e^{x^2+y^2}\right)\text{k},$ find ${\displaystyle {\iint }_S\text{F}·d\text{S}}$ 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 ${\displaystyle {\iint }_S\text{F}·d\text{S}}$ using the divergence theorem if $\text{F}(x,y,z)=2xy\text{i}+3y{e}^z\text{j}+x\text{sin}z\text{k}.$